Index of /Couchman and Jones 2 Unit Book 2

K2

Writing Team Stan Jones Ken Couchman

Keith Slinn Frank Morgan

First published 1983 Reprinted 1984 (twice)

Reprinted 1985 Reprinted 1986 Reprinted 1987 Reprinted 1988 by Shakespeare Head Press

The Education Division of Golden Press Pty. Ltd. Incorporated in N.S.W.

46 Egerton Street,

Silverwater, N.S.W. 2141, Australia

©

1983 S.B. Jones & K.E. Couchman

This book is copyright. No part of it may be reproduced or transmitted without the written permission of the publisher.

ISBN 0 85558 773 3

Typeset in Hong Kong by Asco Trade Typesetting Ltd.

This book completes the 2 Unit course in senior mathematics. For flexibility of programming, Chapter

13 on Probability and Chapter 14 on Sequences and Series overlap with Book 1.

As in Book 1, Practice Papers are placed at intervals throughout the text and a set of 150 Revision Exercises are given at the back of the book for additional practice.

Some proofs in the calculus have again been given in the appendix where students can study them as their mathematical maturity allows.

S.B.J. and K.E.C.

·--·--

~-.--~----I

\ J~r\..1'- I·~j\HiE CLASS C01·..:u1(!\,

-~~

u

~

_de\

--~.J

'vc/

I

-I

I j

i

TI-IIS

BOOK IS THE PROPERTY OF

'

SHOALHAVEN HIGH SCHOOL

--Equally likely outcomes- set notation- range of probability - ways of counting n(S) and

n

(E) - bias - probability trees - comple-mentary events- non mutually exclusive events.

The general term - the arithmetic sequence -sigma notation - geometric sequences -compound interest - superannuation - time payments- limiting sums.

Graphs of quadratic functions - quadratic inequalities - the discriminant of a quadratic equation - sum and product of the roots -maximum and minimum - the sign of the quadratic function - quadratic identities -equations reducible to quadratics.

The second derivative - points of inflexion -turning points- curve sketching- the primi-tive function - application of the primitive function.

Examples of locus - the circle - the parabola - general locus proofs.

The graph and derivative of y

=

ax -

the exponential function y = ex - differentiation of exponential functions - integration - the logarithm function - differentiation and inte-gration -the derivative of y =

ax

-derivative of log8

x -

a value for

e.

Radian measure of angles - length of circular arc - area of a sector - graphs of the trig-onometric functions - graphical solution of equations - differentiation of the trigonometric functions - integration of the trigonometric functions.

The results given by this probability machine are related to Pascal's Triangle.

CHAPTER 13

Probability is the study of unpredictable events and originated in the mid-17th century inspired by the enquiry of gamblers seeking information to help them win at cards and dice. Today probability has many important commercial and scientific applications. Life assurance companies, for example, have tables of life expectancy to help them calculate their premiums, businessmen carry out surveys to help predict the probable market size for a given item. These are attempts to make fairly precise statements about the chance or probability of an event happening.

An important idea in the study of probability is the idea of randomness. Suppose five different

name cards are placed in a hat and one name is drawn at random; what do we mean? We mean that

each name is equally likely to be drawn from the hat and that it is not possible to predict which name

would be drawn.

1. Suppose one card is selected at random from this set of

five cards.

What colour card would most probably be selected? Is the card selected more likely to be a King or a ten?

2. A box contains 5 black balls and 4 red balls. You are blindfolded and asked to draw out one ball from the box. Is this a random choice? What colour ball would you most probably draw?

3. The numbers 1 to 9 are written on separate cards and the cards are turned face down on a table.

One card is chosen at random. Is it more likely to be more than 7 or less than 7? Is it more likely

to be odd or even? Why?

4. Suppose a bag contains blue marbles and red marbles. You are asked to take one marble at random

from the bag and you are told that it will probably be blue. What does that suggest to you about

the marbles in the bag?

When the occurrence of events is equally likely we can state the expected probability of an event occurring by considering the possible outcomes.

Tossing a Coin

Consider the tossing of a coin. There are two possible outcomes, heads or tails, and if it is a fair coin

Rolling a Die

Each ofthe numbers 1 to 6 is equally likely to be on top. There is one chance in 6 ofthrowing the number 4. We say the probability

of throwing a 4 is

i.

What is the probability of throwing a 6?

Drawing a Marble from a Bag

Possible Outcomes

A bag contains 3 red and 2 black marbles. If one marble is drawn at random it is equally likely that any one of the five marbles will be chosen. Consider the probability that the marble chosen is red. Now of the 5 possible outcomes, 3 of them are favourable for the choice of a red marble. We say the

probability of drawing a red marble is~. What is the probability of drawing a black marble?

In general the probability that an event will occur can be given as P(Event) where:

P(E ven -t) _ number of favourable outcomes . . number of posstble outcomes

Further: (i) if there are a favourable outcomes for an event and b unfavourable outcomes and all

outcomes are equally likely then:

the probability of the event occurring = ____!!____b

a+

the probability of the event not occurring = ___!!__b

a+

Note that the sum of these two probabilities is 1 since ____!!____b

+

___!!__b = a

+

bb = 1.

a+

a+

a+

(ii) if an experiment has n equally likely outcomes E_{1 ,} E2 , E3 , . . • , E,. then

P(E1)

+

P(E2 )

+

P(E3 )

+ · · · +

P(E,.) = 1.

1. Toss a coin 50 times and keep a record of the number of heads and tails.

(a) Based on your sample is the probability of tossing a head close to

!

?

(b) Compare your results with other groups.

(c) Now combine the results of ten groups. Based on this larger sample, is the probability of tossing a head close to the expected probability?

2. Roll a die 60 times and record the results using a tally (a) Based on your sample, what is the probability of

throwing a 3?

(b) What would be the theoretical probability of throwing a 3?

(c) Combine the results of ten groups. What is the ratio of the number of times 3 appeared to the total number of tosses? Is this ratio close to the

expected probability of

i?

3

TAILS

1

2 3 4 5 6

7f!./ tfi.J I I

7fl/. I

11/.1.

7tf/.. II

Ill

7111. Ill

3. (a) From a well shuffled pack of 52 playing cards draw a card and record the result with a tally mark for the events listed in the table. Replace the card and reshuffle the pack. Do this 50 times.

(b) Combine your results with other groups and find the experimental probability of each of the out-comes. Compare this with the theoretical prob-ability of each outcome.

BlACI< CARD

DIAMOND

J, a. 1<. A

OF ANY SUIT

7IIJ. 7f/.l. /Ill

7IIJ. Ill

7fll. 7111. I

I. A bag contains 3 black marbles and I white marble. If one marble is drawn out of the bag without looking what is the probability that it is black?

2. A coin is chosen at random from 3 two-cent and 4 five-cent pieces. What is the probability that it is a two-cent piece?

3. From a pack of cards the four aces are turned face down on a table and thoroughly mixed. A girl selects one card. What are her chances of selecting:

(a) the ace of clubs? (b) a red ace?

4. Five pupils: Susan, Robert, Janet, Ian and Gloria write their names on separate cards and put them into a hat. One name is drawn from the hat. What is the probability that:

(a) the name is Janet? (b) the name is a girl's name? (c) the name is a boy's name?

5. A die-cube has 2 faces red, I face white and the other faces blue. What is the probability if the die is rolled on a table that:

(a) a red face would be uppermost? (b) a blue face would be uppermost?

6. If the probability of an event is-!-, about how many times would you expect it to occur in 80 trials?

7. A small deck of cards contains 10 blue and 10 yellow cards. A blue card is drawn from the pack and not returned. If a random draw is then made, which colour is most likely to be selected?

0

/

1 2 3

Passe 4 5 6 Manque

high low

19-36 7 8 9 1-18

10 11 12

13 14 15

Pair Impair

all 16 17 18 all

the the

even 19 20 21 odd

numbers numbers

22 23 24

25 26 27

Noir 28 29 30 Rouge

(black) (red)

31 32 33

34 35 36

'

r

/)'II

~

Colonnes

(Column of 12 numbers)

The slots are numbered 1-36 of which half are red and half are black. They are so chosen that some red numbers are odd and some are even, some are low numbers (1-18) and some are high num-bers (19-36). With only 36 numnum-bers the probability of an odd number is! or 50%. In this case you would have just as much chance of winning as the casino and there would be no profit for the operator. For this reason a zero (0) slot is included on the roulette wheel and the probability is altered slightly in favour of the casino. All winning bets are paid as if there are 36 slots. You can bet on the zero but it does not count for bets on odd or even, red or black and if the marble falls in the zero slot all bets on these are lost.

People indicate their bet by placing tokens on appropriate places on the table. The token can be placed on one number, or between a pair of numbers or on a corner where four numbers meet, in-dicating the choice of any of the four. French words abound in roulette because of its French origin.

Thus a token placed at the end of a row of three numbers is called transversale and at the end of a

group of six numbers is called sixaine and indicates bets on these numbers. Other words are shown

above on the diagram and you will meet some in the exercises. The call for players to place bets is

Faites vos jeux and when the croupier calls Rien ne va plus there are no more bets allowed.

Example:

Carre refers to a group of 4 numbers. What is the probability that one of these numbers will win?

Solution:

P(group of 4) = ₃4_{7 •}

These questions refer to a 37 slot French roulette wheel.

1. What is the probability for a black number (nair)?

2. What is the probability for any even number (pair)?

3. A person choosing colonnes wins if any of the twelve numbers in that column occurs. What is the

probability for this?

4. What is the probability for success of a single number (en plein)?

5. Sixaine refers to a group of six numbers. What is the probability that the marble will end against one of them?

6. What is the chance of success with: (a) cheval (a pair of numbers)?

(b) transversale (a group of 3 numbers)? (c) douzaines (a group of 12 numbers)?

7. Ken put his tokens on the individual numbers 7, 19,20 and 33. What chance has he of success?

8. Mary chose all the odd numbers and, as well, the numbers 6 and 14. What is her probability for success?

9. What is the chance that one of the numbers from 1 to 8 will win?

10. Jenny covered all the low numbers (1 to 18) and also from 31 to 36. What chance did she have of success?

We can apply the notation of sets to the statement of probability in favour of an event. Suppose we wish to find the proba-bility of throwing a number greater than 4 with one roll of an ordinary die. Let S be the set of possible outcomes called the

sample space, then S = {1, 2, 3, 4, 5, 6}.

Let E be the subset of outcomes

favourable to the required event, then E

=

{5, 6}. If P (Event) is the probability of the given event occurring then:

P(Event)

=

n(E)

=

number of elements favourable to event

n(S) number of elements in the sample space

Thus P (number greater than 4) =

%

_ _1 - 3

Example (i):

Sample Space

Event Subset

From a normal pack of 52 cards, one card is selected at random. What is the probability that it is: (a) an ace?

Solution:

(a) Now if E is the set of favourable elements,

n(E)

=

4, since there are 4 aces in the pack and

n(S)

=

52, since there are 52 cards in the pack. Since every element in the sample space is equally likely to be selected

P ( _{ace -}) _ n(E) _n(S)

=

s4z

_ _1_

- 13

Thus the probability of drawing an ace from a deck of 52 cards is /3 •

(b) Now if E is the set of favourable outcomes

n(E) = 13, since there are 13 clubs in the pack.

P (club)

=

n(E)

n(S) _ _u

- 52 _ _!_

- 4

Thus the probability of drawing a club is!.

Example (ii):

A four-digit number is to be formed from the digits 1, 3, 5 and 8 which are printed on cards.

What is the probability that the number will: (a) start with the digit 3?

(b) be odd?

(c) be greater than 5000?

Solution:

(a) We are in this part concerned only with the first digit in the number. It is possible to fill this place with any one of the four given digits. Thus the sample space,

s

= {1, 3, 5, 8}

Now only the digit 3 will give the required event.

.'. E = {3}

P (number starting with 3) = n(E))

n(S

=!

Thus the probability that the number starts with the digit 3 is!.

7

(b) In this part we are concerned only with the last digit (the units digit). It is possible to fill this place with any one of the given digits .

... s

=

{1, 3, 5, 8}

For the number to be odd the last digit must be odd, thus E

=

{1, 3, 5}.

P (odd)

=

~~~~

_i\_ - 4

Thus the probability that the number is odd is

l

(c) The number will be greater than 5000 provided it starts with 5 or 8. Again considering the first place

s

= {1, 3, 5, 8} E = {5, 8}

. n(E)

.. P (greater than 5000) = n(S)

-1. - 4

-1.

- 2

:. The probability that the number is greater than 5000 is

t·

If the probability in favour of event A is {0 and the probability

in favour of event B is~' then event A is more likely to occur

than event B. We could say that there is a high probability

that event A will occur and a low probability that event B will

occur.

The lowest probability that can ever be quoted is zero, and this when an event can never occur, e.g., selecting a 5 cent coin from a box containing only 10 cent coins.

The highest probability that can ever be quoted in favour of an event is 1 and this only when the event is certain, e.g., drawing a red marble from a box containing only red marbles.

Thus the range of probability for an event is from 0 to 1 and to quote any number outside this range is meaningless.

1. If a die is rolled once, what is the probability of throwing: (a) 6?

(b) an even number? (c) a number less than 3?

1

5

1

1

2

0

5

8

3

1

8

Probability

...,.__ CERTAIN ...,.__ LIKELY

...,.__ EVEN CHANCE

...,.__ UNLIKELY ...,.__.. IMPOSSIBLE

2. The probability that a car driver will be stranded by mechanical failure in 15 000 km of driving is 1

3

0 • What is the probability that the driver will have trouble free motoring for the 15 000 km?

3. At a school assembly 62% of the students said they were in favour of a new design for the school

badge. If the headmaster asked a student at random about the proposed design, what is the

4. In a raffle 40 tickets are sold and there is one prize. What is the probability that someone buying five tickets wins the prize?

5. Give the probability of each of the following selections occurring at a single random draw from a pack of 52 playing cards:

(a) the ace of clubs (b) a seven of any suit (c) a black Jack

(d) a court card (Jack, Queen, King, Ace) (e) a red card

(f) either the king of spades or the queen of diamonds.

6. A shelf contains 9 different books, arranged in any order between a wall and a book-end. Three are yellow, 5 are blue, and one is green. What is the probability that:

(a) a yellow book is in the middle position? (b) a blue book is against the wall?

(c) the green book is beside the book-end?

7. The numbers I to 20 are written on twenty cards. If a card is chosen at random what is the prob-ability that :

(a) the number is a multiple of 5? (b) the number is even?

(c) the number is odd?

(d) the number will contain the digit 7? (e) the number will contain the digit 2?. (f) the number will contain the digit I?

8. A bag contains 3 blue, 5 red, 4 green and 2 white balls. Find the chance of drawing at random one ball which is:

9.

(a) red

(b) either blue or green (c) not white.

The organisers of a local procession which is held every year on the first Saturday in June have available a record of the weather over the past fifty years on the procession day.

Based on the record given in the table what is the probability that this year's procession will be:

(a) held on a sunny day with no cloud? (b) held on a rainy day?

4

/

10

/

15

/

/

21

/

10. A number is formed by using all five digits 2, 3, 4, 5, 6. What is the probability that the number:

(a) starts with 4? (d) is greater than 30 000?

(b) is even? (e) is divisible by 3?

(c) is odd?

11. Joel and Debbie are each drawing one card at random from a pack of ten cards numbered 1 to 10. Joel draws a "6" which is replaced and the pack shuffled. What is the probability that Debbie will draw a higher card?

12. From a set of 18 discs numbered from 1 to 18, one is drawn at random. What is the probability that the number on the disc is:

(a) a multiple of 5 (b) a multiple of 3 (c) a multiple of 4

13. A bag contains 5 white and 8 green marbles.

(d) a multiple of either 4 or 5 (e) a multiple of either 3 or 5 (f) a multiple of either 3 or 4.

(a) What is the probability of drawing at random, a white marble from the bag?

(b) If a white marble is drawn first and not replaced, what is the probability of picking up a white marble at the second draw?

(c) Suppose white marbles have been selected at the first and second draws and not replaced, what is the probability of picking a green marble at the third draw?

In some problems it is helpful to draw a diagram or use some other systematic procedure in order to determine the elements in the sample space.

TWO DICE ROLLED SIMULTANEOUSLY

The diagram shows the sample space (i.e. possible outcome) when two dice are rolled together. We have shown one die white and one die red to indicate, for example, that there are two distinct

sample points whose sum is 3, i.e., 2 on the white and 1 on the red, and vice versa.

Example:

36

SAMPLE POINTS

What is the probability of throwing a total of five in one roll of a pair of dice?

Solution:

We can see that all the sample points whose sum is five lie in one line parallel to a diagonal. If we write the number on the white die first, the sample points can be written as ordered pairs. Now E

= {

(1, 4), (2, 3), (3, 2), (4, 1)} thus n(E)

=

4, and from the diagram n(S)

=

36.

. . n(E)

.. P(f1ve) = n(S)

=

3~

_ _1 - 9

1. A pair of dice is thrown. Use the diagram to count the number of sample points favourable to the events listed below and hence find the probability of the event in each case.

(a) a total of 7 (b) a double six (c) any double (d) a total of 10

(e) either a score of 7 or a score of 11

(f) a score greater than 6

(g) an even score

(h) an odd score also greater than 4

(i) at least one six on the uppermost face of a die

(j) at least one three on the uppermost face of a die.

2. Which would give the better chance; rolling one die or rolling two dice, if you wanted to throw a score of, (a) 6? (b) 2?

3. A game is played so that a player can start only if he throws a double or a total of six, using two dice. What is the chance of a player's starting on the first throw?

Craps is a game of chance played by rolling two dice. The total made by the two dice is what matters. This is a game played in all big casinos.

WIN LINE IN CRAPS

This is an even money wager and there are three rules:

1. If, when the two dice are rolled, the total is 7 or 11, the player wins.

2. If a total of 2, 3 or 12 shows, the player loses immediately.

3. If any other total such as 5, shows; this is called the player's point. The player keeps on rolling the dice. He wins if this total (his point) appears before a 7 and he loses if a 7 appears first.

1. Complete the following table showing the number of ways various totals can be obtained with

two dice.

TOTAL OF 2 DICE 2 3 4 5 6 7 8 9 10 11 12 All outcomes

Number of ways 36

2. What is the chance of losing on the first throw, that is by scoring a total of 2 or 3 or 12?

3. If on the first throw a total of 5 appears is it more likely to gain another 5 before a 7?

4. Is it more likely to win or lose on the first throw?

5. What is the probability that the game will end one way or the other with the first throw?

6. What is the probability that the game will not end on the first throw?

7. If the first throw is 10 what is the probability that the next throw will be a 10? (Hint: The dice can't remember what their first throw was! This question is simply asking what the chance is of throwing a 10.)

8. If none of the numbers 7, 11, 2, 3 or 12 occurs first up what is the next best total to get? Mention

two if they are equally desirable.

9. Write all the totals in the order they would be preferred. For example a 7 or an 11 are the most

desirable totals. What is the remaining order?

A tree diagram can be used to find the elements in the sample space.

Example (i):

The numbers 2, 3, 4 and 5 are written on separate cards. One card is drawn at random to give the tens digit of a two digit number and then one of the remaining cards is also drawn at random to give the units digit. What is the probability that the number formed will be divisible by 6?

Solution:

From the tree diagram we can see that the set of two digit

numbers in the sample space contains 4 x 3 = 12 elements.

The numbers in the sample space which are divisible by 6 have been ticked.

Thus the probability that the number formed is divisible by 6 is given by:

P(number divisible by 6)

=

~~;?

3 12 1 4

3 23

2 ~ 4 24 5 25 2 32

3 4 34

5 35 2 42

4 3 43

5 45 2 52

5 3 53

4 54

The main value of the tree diagram is that it lists every possible element in the sample space. In this case every two digit number that could be formed is found by following all possible branches on the tree diagram.

Another value of the tree diagram is that it illustrates an important principle of counting possibilities

known as the product rule.

If a selection can be made in r different ways and if a second selection can be made ins different

ways then the two selections can be made in sucession in r x s different ways.

Example (ii):

Solution:

Draw a tree diagram to obtain the sample space.

H H

T H

H T

T H H

T T

H T

T

The three sample points ticked give the required event .

.'. P(two heads and one tail) =

:~~~

_.J.

- 8

.·. The probability of throwing two heads and a tail is

i.

HHH

HHT

HTH

HTT THH

THT

TTH

TTT

Note: If three coins are tossed together the sample space is identical with that in the tree diagram

as the headings 1st toss, 2nd toss, 3rd toss, can be replaced by 1st coin, 2nd coin and 3rd coin.

Example (iii):

A simple poker machine has only two wheels. On one of them there are the numbers 1, 2, 3 and 4. On the other one the letters A, A, B and C.

(a) How many arrangements can show on the dial (including repetitions)?

(b) In how many ways can the machine stop with the first wheel on a 3 and the second wheel showing anA?

(c) What is the probability that the arrangement 3-A will appear?

Solution:

We draw a tree diagram:

A 1A

A 1A

8 18

c 1C

2A (a) Altogether 16 arrangements are 2A possible (not all different). 2

28 (b) 3-A occurs twice.

--c 2C _{(c) P(3-A)}

=

_n(E)

3A n(S)

A 3A _-_ _.1_ ₁₆

3

38 -_1_ 8

3C

~/A 4A

4A 4

48

Where necessary draw a tree diagram or use some systematic procedure for finding the sample space in each of the following problems.

1. A set of three cards numbered 1, 2 and 3 are placed in a hat.

(a) How many different two-digit numbers would it be possible to form by selecting two cards in succession and placing them in order?

(b) Use a tree diagram to list all the possible numbers.

(c) What is the probability that any two-digit number formed in this way is even?

2. Two white balls and one black ball are placed in a bag. Two balls are selected at random. Find the probability of choosing:

0

00

(a) 2 white balls.

₀

(b) 1 white and 1 black ball.

•

oe

0

00

0

•

oe

0

eo

•

0

eo

3. A coin is tossed three times. Use a tree diagram to find the probability in favour of the event: (a) one head and two tails in any order

(b) three heads or three tails (c) at least one tail.

4. A tennis team of four players A, B, C and D select a captain and vice-captain.

(a) List all the possibilities.

(b) What is the probability that player A will be either captain or vice-captain?

5. Four Jacks are taken from a pack of cards and placed face down. Two of these cards are chosen at random. Use the tree diagram given to help find the probability of choosing:

(a) 2 black Jacks. (b) at least 1 black Jack.

SPADE

CLUB

HEART

CLUB HEART DIAMOND SPADE HEART DIAMOND SPADE CLUB DIAMOND SPADE DIAMOND CLUB

6. A coin is tossed 4 times. What is the probability of 3 heads and a tail in any order?

7. Two coins are tossed together. What is the probability of: (a) 2tails?

(b) 1 head and 1 tail?

8. If you toss a coin and throw a die, what is the probability of getting: (a) a head and a 4?

(b) a tail and an odd number? (c) a head and a number less than 3?

9. There are two roads m and n between towns A and B.

There are three roads x, y and z between towns Band

C. If the path is chosen at random from A to C, what

is the probability that a traveller will journey over

roads m andy? A

n

10. A two-digit number is to be formed from the set of digits {3, 4, 5, 6}.

m

(a) If any digit may be repeated how many two-digit numbers can be formed? (b) What is the probability that the number formed is divisible by 5?

(c) What is the probability that the number formed has both digits the same?

11. Use the tree diagram given to find the sample space for the composition of a family of 3 children. For such a family what is the probability of:

(a) 3 girls?

(b) 2 boys and 1 girl?

(c) the eldest child being a girl? (d) the youngest child being a boy?

B

G

/

X

z

B

/ B

G

B G

G

B B

G

B G

G

12. From a set of 4 cards, consisting of the Kings from a normal pack, 3 are chosen at random. Find the probability that the two red Kings would be included in the 3 cards chosen.

13. Cards in a set are numbered 1, 2, 3, 4 and 5. If two cards are drawn at random: (a) write down all the possible two-digit numbers that can be formed

(b) what is the probability that the number formed is odd?

14. Three yellow cards are numbered 1, 2 and 3. Three red cards are also numbered 1, 2 and 3. -If a yellow and a red card are selected at random, what is the probability that the cards selected: (a) are the yellow 2 and the red 3?

(b) give a total of 5?

(c) are the yellow 3 and the red 3? (d) give a total of 2?

(e) have a total greater than 3?

15. A drawer contains 2 black and 2 red ball-point pens, and nothing else. If the pens are identical in

size and shape, what is the probability if two are chosen in the dark, that they will both be red?

16. Three books are on a shelf. One book has a red cover and the other two have blue covers. If two

books are selected by chance what is the probability that the red book is included in the selection?

17. Five golf balls are in a pouch. Four have red numbers and one has a blue number printed on it.

If two are selected at random, what is the probability that the one with the blue number is in the

pair selected?

18. (a) A poker machine has 4 wheels and 20 symbols on each. How many arrangements are possible? (b) Suppose it is possible for a certain winning combination to occur in 4 ways, what is the

prob-ability of this occurring?

19. (a) A poker machine with 3 wheels has 4 Aces on the first wheel, 2 Kings on the second wheel and 3 Jacks on the third. In how many ways can the machine stop to show Ace, King, Jack? (b) If each of the three wheels has 10 symbols, then how many total positions are possible and

what is the probability for Ace, King, Jack?

At times outcomes of experiments are not equally likely. Sometimes this is due to bias. If the nature

of the bias is known then allowance can be made for it in the sample space.

Example:

A die is biased so that to throw a six is twice as likely as any other number. Find the probability of throwing:

(a) a three (b) a six.

Solution:

The six is given extra weight in the sample space by counting it twice. Thus S = {1, 2, 3, 4, 5, 6, 6}

(a) P (throwing a three) =

:~~j

_ _i

(b) P (throwing a six)

- 7

n(E)

n(S)

_l_

- 7

1. A die is biased so that to throw a 2 is twice as likely as any other number. Find the probability of throwing:

(a) a six (b) a two.

2. The 38 slots on an American roulette wheel are marked 1-36, 0, 00. If the wheel is biased so that 00 is twice as likely as any other slot to pick up the ball, find the probability of a player winning if he bets on number 7.

3. A coin is so weighted that over many trials the ratio of heads to tails will be 5 to 3. Find the probability, at a single trial, of throwing a tail.

In some questions on probability, basic tree diagrams become very large and unwieldy. It is then a

useful economy to use a probability tree.

Examples for study

Example (i):

A bag contains 3 white and 4 brown balls. Two balls are drawn at random, the first not being replaced

(a) The probability of White at the first draw is ~. Now given the condition that the first draw resulted in White, the prob-ability of White at the second draw is ~' because there are now six balls in the bag of which two are White.

The probability of getting White at each draw can be cal-culated by multiplying the probabilities along the coloured branch.

P(White, White)

=

~ x ~

=

fz

=

~

2

6

w

This rule can be verified by noting that these are 3 ways of picking a white ball at the first draw

and 2 ways at the second draw. Thus by the product rule the sequence White, White can be chosen

in 3 x 2

=

6 ways. There are 7 possible outcomes of the first draw and 6 for the second draw,

giving 7 x 6 = 42 possible outcomes for the two draws in sequence.

This gives a probability for White, White of ,f2 which corresponds with the result obtained by

multiplying the probabilities along the branch giving White, White.

The product rule for a probability tree states that the probability of an event is the product of the probabilities along the branch which yields the event.

(b) Now the probability of each outcome can be easily calculated by using the product rule above.

P(White, Brown)

=

~ X ~

=

g

= t

S · l

P(White, White) =

l

x ~

=

,f2

=

~

t

P(Brown, White)

=

4

x ~

=

!~

= t

urn ts

P(Brown, Brown)=

4

X~=

g

=

t

Notice if the sum of the probabilities along the separate branches is 1, we have a check on our work.

(c) If the required event is drawing a white ball and a brown ball in any order, we note that this can be done in two ways (White, Brown) or (Brown, White). Thus to find the probability of this event the two probabilities concerned are added.

P(White and Brown)

=

P(White, Brown)

+

P(Brown, White)

=t+t

- ± - 7

The addition rule for a probability tree states that the probability of a set of mutually ex-clusive events is the sum of the probabilities of corresponding branches.

Example (ii):

If a coin is tossed and a die thrown, find the probability that a head and a number greater than 2 results.

Solution:

The coloured branch gives the required event. Thus the prob-ability of the event is the product of the probabilities along this branch.

P (head and > 2) =

!

x ~

=

ti

_.!_

- 3

Example (iii):

There are nine cards. The numeral 4 is printed on three of the cards, the numeral 5 on two of them and the numeral 7 on four of them. One card is drawn at random and not replaced, giving the tens digit of a number. Then another card is drawn to give the units digit. Find the probability that the number formed:

(a) is 44

(b) contains a repeated digit.

Solution:

Using the probability tree diagram. (a) P(44) = ~ X

i

_ _§_

- 72 _ _L - 12

(b) The branches giving repeated digits are shown in colour. The probabilities along each branch must be found by multiplication and then P(44), P(55) and P(77) must be added.

P(repeated digits) = P(44)

+

P(55)

+

P(77)

=

(~ X

i)

+

(~ X

i)

+

(~ X ~)

= 762

+ /2

+

n

=

~~

=

ls

1. In Example (iii) above, find the probability that the number formed,

(a) is 47 (c) has a digit sum of 11

(b) is 74 (d) contains the digit 5 at least once.

2. A box contains 3 red and 2 black marbles. Two are drawn at random, the first not being replaced before the second draw. Find the probability that:

(a) 2 red are drawn

(b) a black is drawn followed by a red (c) the selection contains a red and a black.

3. (a) A die is thrown three times. Find the prob-ability of:

(i) three sixes (ii) no sixes (iii) two sixes

(iv) at least two sixes.

(b) Would the probabilities above be altered if three dice were thrown together?

4. To win a golf tournament a professional is faced with two putts, one on the 17th green which he has an even chance to sink, and one on the 18th green that he would sink 3 times out of 4. What is his chance of:

(a) winning the tournament? (b) missing both putts?

5. A certain canteen plans its lunch menu in 6 day units. On two days potato pie comprises the first course and on 3 days fruit salad is served as dessert. What is the probability that a given meal chosen at random includes:

(a) neither? (b) both?

6. The probability that a certain missile will blow up its target is~. If two missiles are launched in quick succession, what is the probability that the target will escape?

7. What is the probability of throwing six on a die four times in succession?

8. The probability that a car driving on sealed roads will have a puncture in 25 000 km is

l

What is the probability that this car does not have a puncture during the first 100 000 km? A new set of tyres is fitted every 25 000 km.

9. If

i

of the boys in a certain class carry a briefcase, and

i

are out of uniform, find the probability that a boy selected by lot, would be in uniform and carrying a briefcase.

10. Of7 equally matched toy racing cars, 2 are white, 4 are red, and one is green. Find the probability that if two races are run :

(a) both will be won by a red car

(b) both will be won by a car of the same colour

(c) the first race will be won by a red car and the second by a white car (d) that one race will be won by a green car and one by a red car.

11. The probability that a tail-end batsman will lose his wicket on any ball is 0·1. The probability that he survives the first ball of an over is therefore 0·9. The probability that he survives the first and second balls is given by the product rule as 0·9 x 0·9. What is the probability that he survives: (a) the first 3 balls?

(b) a complete 6 ball over?

12. What is the probability of drawing the ace of hearts on two successive occasions from a pack of

52 playing cards with replacement and shuffling between each draw?

13. A television set makes a whistling noise 5 times out of 6 when it is switched on. When the tech-nician comes to repair the set what is the probability that it does not make the whistling sound when switched on:

(a) once? (b) twice? (c) three times?

14. When an egg is broken from the shell into a frying pan the probability that the yoke breaks is said to be 1 in 10. If three eggs are fried, what is the probability that no yokes will be broken?

15. On a three-wheel poker machine these are 10 symbols on each wheel. There are 5 lemons on the first wheel, 3 on the second and 2 on the last.

(a) What is the probability that 3 lemons will show on the centre line?

(b) What is the probability that the two outside wheels will show a lemon with something dif-ferent on the middle wheel?

In a given experiment if E is the subset of outcomes favourable to an event then

E,

the complement of E, represents the subset of outcomes not favourable to the event and P(E) is the probability that the complementary event occurs. The event

E

is often referred to as not E.

E and

E

together form the total sample space thus P(E)

+

P(E) = 1

or P(E)

=

1 - P(E)

This result can be useful in finding probabilities where the words at least are used.

Example (i):

What is the probability that in a family of five children there is at least one boy.

Solution:

The probability of no boys =

t

x

t

x

t

x

t

x

t

_ __!_ - 32

The complement of no boys is at least one boy.

Hence the probability of at least one boy = 1 -

l

2

_ _li - 32

Example (ii):

In three throws of a die, what is the probability of at least one six?

Solution:

The probability of no six in three throws

=

i

x

i

x

i

=iU

The complement of no six is at least one six.

Hence the probability of at least one six

=

1 -

iU

- 91 - 216'

1. If a coin is tossed three times what is the probability of getting at least one head?

2. What is the probability that in a family of four children there is at least one girl?

3. Two dice are thrown. What is the probability of throwing at least one six?

4. (a) In four throws of a die what is the probability of getting at least one six?

(b) If five dice are thrown, what is the probability of throwing at least one six? Hint: The calculations when throwing five dice can be done in the same way as for throwing one die five times.

5. The four Aces from a pack of cards are laid face down. One is selected at random and replaced. In two such selections what is the probability of drawing the Ace of spades at least once?

6. There are two boxes A and B. Box A contains 3 white balls and 2 black balls. Box B contains 4 white balls and 3 black balls. One ball is taken at random from each box. What is the probability that at least one of the balls is white?

7. A die is thrown and a coin is tossed, what is the probability (a) that the die shows six and the coin shows a head? (b) of at least one of the events in part (a) occurring?

8. In a mixture of brown and white pebbles in a gravel, brown and white pebbles occur in the ratio of 2: 3. Find the probability that if three pebbles are chosen from the mixture, at least one is white.

9. On a three wheel poker machine there are 20 symbols on each wheel. There are 5 cherries on the first wheel, 3 on the second and 2 on the third. What is the probability that:

(a) 3 cherries will show across the centre line? (b) at least one cherry will show on the centre line?

Consider the following example.

From a pack of cards numbered 1 to 10, one card is drawn at random. What is the probability that the number on the card is :

(a) less than 5 or divisible by 7? (b) less than 5 or divisible by 2?

Solution:

(a) Let A be the event of drawing a number less

than 5 and B be the event of drawing a

num-ber divisible by 7. The events A and B are

mutually exclusive; we cannot select a num-ber less than 5 and a number divisible by 7 at the same time.

We require A or B and denote this by

A u B (called the sum of A and B). Now

from the diagram.

P(A u B)

=

P(A)

+

P(B)

=

140

+

lo

_.!_

- 2

(b) Let A be the event of drawing a number less

than 5 and B be the event of drawing

anum-ber divisible by 2.

The events A and B are not mutually

exclu-sive; drawing a number less than 5 does not exclude the possibility of drawing a number divisible by 2.

Since A and B are not mutually exclusive

events, A u B contains the points which are

common to A and B; these are the points

An B.

We want P(A u B), but if we add P(A)

and P(B) the points of P(A n B) will have their probabilities counted twice.

Thus P(A u B)

=

P(A)

+

P(B) - P(A n B)

=

1~

+

15

o - /o

=

2o

s

s

6

8

10

5 9

A B

6 7

3 10

9

5

Note: In probability theory A n B is called the intersection or product of the events A and B. It

is written as AB and denotes the event that both A and B occur.

1. If A and B are two events in a random experiment

P(A u B)

=

P(A)

+

P(B) - P(AB)

2. In the special case where A and B are mutually exclusive events P(AB) = 0.

Hence P(A u B) = P(A)

+

P(B).

Example:

One card is selected at random from a pack of 52 playing cards. What is the probability that it is either a red card or a ten?

Solution:

A is the event of a red card drawn; P(A) = ;~

B is the event of a ten drawn; P(B) = 5

i

A and B are not mutually exclusive events as the event AB is a red ten drawn; P(AB) =

l

2 P(A u B)

=

P(A)

+

P(B) - P(AB)

= ~~

+

542 -

l2

=~

=

23

Note: In solving problems the student can work from the above formula or work from a counting diagram such as a Venn diagram or a simple tree diagram. The important thing is to recognise a

situation where the events are not mutually exclusive being careful not to count twice those elements

in the intersection.

In the following exercise some events are mutually exclusive and some are not.

1. From a set of cards numbered 1 to 12, one is selected at random. What is the probability that the number on the card is :

(a) less than 5 or divisible by 6? (b) less than 5 or divisible by 4?

2. A card is selected at random from a pack of 52 playing cards. What is the probability that it is:

(a) a black ten or an Ace? (c) a club or an Ace?

(b) a black card or an Ace? (d) a club or a court card?

3. If two dice are thrown what is the probability of: (a) a double or a total of 8?

(b) a double or a total of 9?

4. A bag contains 3 white marbles, 4 yellow marbles and 2 red marbles. A marble is drawn at random, what is the probability that it is either white or yellow?

5. From the integers 1 to 50 an integer is chosen at random. What is the probability that it is:

(a) divisible by 4 and 6? (b) divisible by 4 or 6?

6. Two dice are thrown. What is the probability of obtaining a total which is either even or greater than 7?

7. From a set of 30 discs numbered 1 to 30, one disc is selected at random. What is the probability

that the number is: (a) a multiple of 5? (b) amultipleof7? (c) a multiple of 3?

(d) a multiple of either 5 or 7? (e) a multiple of either 5 or 3? (f) a multiple of either 3 or 7?

8. (a) If A and Bare mutually exclusive events and P(A)

=

₁5

2 and P(B)

=

132 , find P(A u B).

(b) If A and Bare not mutually exclusive events and P(A) = {0 , P(B) = 240 and P(AB) =

l

0 ,

find P(A u B).

9. From the integers 1 to 11, one is chosen at random. What is the probability that it is less than 9 or divisible by 4?

10. From the numbers 4, 5, 6, 7, 8, 9; what is the probability of selecting at random an odd number or a number less than 7?

A computer can be programmed to give the monthly instalments on loans which are subject to reducible interest. The calculations involve geometric series.

CHAPTER

14

If a set of numbers is written down in order according to some rule or pattern, the set of numbers is

called a sequence and each number is called a term of the sequence. In some sequences the pattern

can readily be seen and this allows us to predict subsequent terms in the sequence.

For example consider the sequence 1, 3, 6, 10, 15, 21, 28, .... The difference between successive

terms increases by 1 each time, so the sequence may be continued indefinitely. The next three terms in the sequence are 36, 45 and 55.

Discover the pattern in each of the following sequences and use this knowledge to write down the next three terms of the sequence.

1. 3, 7, 11, 15, 19, - , _ , - 7. 31, 24, 17, 10, _, _ ,

-2. 3, 4, 6, 9, 13, 18, - , _ , - 8. 17, 8, 1, 10, _, _ ,

-3. 19, 16, 13, 10, 7,_,_,_ 9. 1, -2,4, -8, 16, -32,_,_,_

4. 3,6, 12,24,48,_,_,_ 10. 1, 8, 27, 64, _, ,

-5. 1, 4, 9, 16, 25, _, _ , - 11. 1, 1, 2, 3, 5, 8, 13, _, ,

-6. 2, 6, 18, 54, 162, _, _ , - 12. 4, 3, 7, 6, 10, 9, 13, 12, _, ,

-It is possible to indicate a sequence by giving its general term. The symbol T,, is used to represent

the general term or nth term of a sequence.

For example if T,,

=

3n

+

2, then the first term is found by substituting 1 for n in this expression and so on for the other terms:

Thus T1

=

3 x 1

+

2

=

5

T2 =3x2+2=8

T3=3X 3+2=11 T4 = 3 X 4 + 2 = 14 giving the sequence 5, 8, 11, 14, ....

We could say this is the sequence whose general term is 311

+

2.

1. Find the first four terms of the following sequences whose nth terms are given.

(a) T,, = 211 (d) T,, = 4n - 2 (g) _{T,, = 11 + 2} 11 (j) T,, = 2"-1

(b) T,, = 3(11 - 1) (e) _T,, = 11 + 6 (h) T11 = 11

3 _{(k) T} = 11 + 1

n ₁₁2

(c) T,,=11-4 (f) _{T,, = 211} 1 (i) T,, = 12 - 3n (1) T,, = 3"

2. Write down the first six terms of the sequence given T,, = ~(n

+

1). Do you recognise this

3. If n is an integer, (a) is 2n odd or even? (b) is 2n

+

5 odd or even?

(c) What is the simplest general statement of an odd number?

4. What is the value of ( -l)n when (a) n is even? (b) n is odd?

5. Find the first three terms of the following sequences whose nth terms are given. (a) T,,

= (

-1)n(2n

+

3) (c) T,,

= (

-l)n+13n

(b) T,, = ( -l)n! (d) T,, = ( -1)nn2

•

n

6. The nth term of a sequence is given by T,, = 5n - 23. (a) How many terms of the sequence are negative? (b) Is 32 a term of the sequence? If so, which term?

(c) What is the difference between successive terms of this sequence?

7. Which term of the sequence T,, = 3n

+

7 is 34? 8. If T,, = 35 - 3n

(a) How many terms are positive? (b) Is 16 a term of the sequence?

(c) What is the first term? How could you use the first term to find the second?

An arithmetic sequence or arithmetic progression is a sequence in which each term after the first is formed by adding a constant number to the preceding term.

The sequence 6, 10, 14, 18, 22, 26, ... for example, progresses by adding the constant number 4. This sequence is an arithmetic sequence and the constant number added is called the common difference of the sequence.

Examples of arithmetic sequences are :

8, 10, 12, 14, 16, . . . common difference 2 15, 12, 9, 6, 3, . . . common difference -3 4, 51, 6, 71, 9, . . . common difference 11.

We note from the above that the common difference may be positive or negative.

Example:

Find the common difference of the arithmetic progression 17, 15, 13, 11, ... and write the next two terms of the sequence.

Solution:

Any term subtracted from the term that follows it gives the common difference d . . ·. Common difference

=

d

=

15 - 17

=

-2

. ·. The next two terms are 9 and 7.

1. Find the common difference in each of the following arithmetic sequences and then write the next two terms.

(a) 1, 3, 5, 7, 9, .. . (f) 5, 12, 19, 26, .. . (b) 18, 16, 14, 12, .. . (g) -5, -1, 3, 7, .. .

(c) 12, 91, 7, 41, .. . (h) 4, 4!, 51, 6!, .. .

(d) 60, 70, 80, 90, .. . (i)

J3, JU,

J27, .. .

(e) X, X

+

3, X

+

6, X

+

9, ... (j) 4x, 7x, lOx, ...

2. Determine whether or not the following numbers form arithmetic sequences.

(a) 15, 23, 31 (c) -5, -1,4

(b)

2!, 3i, 5

(d) 3·7, 4·1, 4·5

3. Write down the first three terms of an arithmetic sequence given that (a) the first term is 6 and the common difference is 7.

(b) the first term is 8 and the common difference is - 3.

(c) the first term is -4 and the common difference is

It,

4. If a, b, c forms an arithmetic sequence, show that b

=

a ; c.

5. The first three terms of an arithmetic sequence are 7, x, 35. Find x.

6. If a, b, c, d, e are in arithmetic sequence, show that a

+

e

=

b

+

d

=

2c.

The nth term of an arithmetic sequence may be found by letting a represent the first term and d the

common difference.

The sequence is then: a, a+ d,a

+

2d,a +3d, a+ 4~ ....

The nth term of this sequence may be obtained by examining the following table:

First term Second term Third term Fourth term Fifth term

...

nth term

Tt Tz T3 T4 Ts

...

T,,

a a+d a+ 2d a+ 3d a+ 4d

...

a+ (n- l)d

Can you see the pattern? The coefficient of din each term is one less than the number of the term.

Example (i):

The nth term of the arithmetic sequence

a, a

+

d, a

+

2d, a

+

3d, ... is given by T,, = a

+

(n - l)d.

Find the twentieth term of the sequence 5, 8, 11, 14, ....

Solution:

Here a

=

5 and d

=

3

Using T,, = a

+

(n - l)d

T20 = 5

+

19 x 3

= 62

Therefore the twentieth term is 62.

Example (ii):

Which term of the arithmetic sequence 15, 11, 7, ... is - 33?

Solution:

T,, = a

+

(n - l)d

-33

=

15

+

(n - 1) x ( -4)

-33 = 15- 4n

+

4

4n = 52

n = 13

Therefore -33 is the thirteenth term.

Example (iii):

Find the first term and the common difference of an arithmetic sequence whose fifth term is 31 and whose twelfth term is 73.

Solution:

T₅ =a+ 4d = 31 T12 = a

+

lld = 73

.'. 7d = 42

.·. d = 6 and a= 7

Therefore the first term is 7 and the common difference is 6.

Example (iv):

Insert four numbers between 8 and 23 such that the six numbers form an arithmetic sequence.

Solution:

Since we are inserting four numbers between 8 and 23 then 8 is the first term and 23 is the sixth term of the arithmetic sequence.

Using T,,

=

a

+

(n - l)d

23

=

8

+

5d

.'. d

=

3

Hence the four numbers are 11, 14, 17, 20 and the sequence is 8, 11, 14, 17, 20, 23.

1. Find the term indicated in each of the following arithmetic progressions.

(a) lOth term of 7, 11, 15, 19, . . . (e) 13th term of 4, 61, 9, .. .

(b) 16th term of 18, 15, 12, . . . (f) 55th term of 5, 7, 9, .. .

(c) 9th term of -9, -4, 1, . . . (g) 40th term of -4, -8, -12, ...

(d) 20th term of 11, 16, 21, . . . (h) 8th term of 5

+

2b, 5

+

5b, 5

+

8b, ...

2. What is the nth term of the sequence (a) 1, 3, 5, 7, 9, ... ?

(b) 2, 4, 6, 8, 10, ... ?

3. What is the general term, T,, of the sequence 7, 14, 21, 28, ... ?

4. Find the nth term of the arithmetic sequence 20, 17, 14, ... .

5. The nth term of an arithmetic sequence is given by T,,

=

4n

+

8. Write the first four terms and

the common difference.

6. The general term of an arithmetic progression is 2n - 1. Write the first three terms and the

60th term.

7. Which term of the sequence 7, 12, 17, ... is 372?

8. Is 279 a term in the arithmetic sequence 13, 17, 21, ... ?

9. Find how many terms there are in the sequence 9, 12, 15, ... , 195.

10. If the nth term of an arithmetic sequence is given by T,,

=

13 - 2n, find the first three terms, the

common difference and the 1 OOth term.

11. If the first term of an arithmetic sequence is 9 and the fourth term is 27, find the common difference.

12. In an arithmetic progression T7 = 20 and T13 = 38, find the first term and the common difference.

13. In an arithmetic sequence T10

=

5 and T17

=

54, find the first three terms.

14. Find T₅₀ of an arithmetic sequence in which T5

=

13 and T9

=

25.

15. Find the number of terms in an arithmetic sequence with a= 5, d

=

2 and the last term 43.

16. The fourth term of an arithmetic sequence is 116 and the common difference is 9. Find (a) the first term (b) the 30th term.

17. The first three terms of a sequence are 3, 7, 11, .... What is the first term to exceed 200?

18. Write down the numbers 24 and 44, then insert three numbers between them so as to give five numbers in arithmetic progression.

19. Insert six numbers between 43 and -6 such that the eight numbers form an arithmetic sequence.

20. The first three terms of an arithmetic sequence are 48, 41, 34.

(a) Write down a formula for the nth term.

(b) If the last term of the sequence is -29, how many terms are there in the sequence?

21. After starting full-time work a girl saves $15 in the first week, $19 in the second week, $23 in the third week and continues to increase her savings each week by the same amount until the twelfth week. How much is she then saving each week?

22. The temperature of the water in a boiler is rising at a constant rate. Readings taken every 15

minutes are as follows 25°C, 28°C, 31

oc

and so on. The last reading taken was 91 °C. How many

readings were taken in all?

23. The sum of the first three terms of an arithmetic sequence is 24 and the sum of the next three terms is 51. Find the first term and the common difference of this sequence.

24 . I f x = - -2ab s ow t at -, -, h h 1 1 1 -b are m . ant metlc progressiOn. . h . .

a+ b a x

Let us consider the sum of the arithmetic sequence

a

+

(a

+

d)

+

(a

+

2d)

+ · · · +

(l - 2d)

+

(! - d)

+

l

where a is the first term, dis the common difference and lis the nth term. Note that the terms next

to the last will be (l - d), (! - 2d), etc. Thus if S11 stands for the sum of n terms of this sequence

sll

= a

+

(a

+

d)

+

(a

+

2d)

+ . . . +

(l - 2d)

+

(! - d)

+

l

also S11

=

l

+

(!- d)

+

(l- 2d)

+ · · · +

(a

+

2d)

+

(a

+

d) +a The sum does not change

when the order of the terms is reversed. Adding the two series, term by term, we have

2S11 = (a

+

!)

+

(a

+

!)

+

(a

+

1)

+

+

(a

+

!)

+

(a

+

!)

+

(a

+

1)

that is 2S,, = n(a

+

!)

. n

.. sll

= 2(a

+

!)

The sum of n terms of an arithmetic sequence, given the first term and the last term is :

n

sll

=

2(a

+

!)

Note also that since 1 is the nth term, l = a

+

(n - l)d.

n

Then S11 =

l

[a

+

a

+

(n - 1 )d]

.·. sn

=

~

[2a

+

(n - l)d]

The sum of n terms of an arithmetic sequence, given the first term and the common difference is:

n

Example (i):

Find the sum of 20 terms of the series 7

+

12

+

17

+

22

+

Solution:

Here a= 7 and d = 5.

Using S"

=

~[2a

+

(n - l)d]

20

s2o

=

2

[14

+

19 x 5]

=

10 X 109

= 1090.

Example (ii):

Find the sum of the series -15- 8 - 1

+

6

+

13

+

Solution:

Here a

=

-15, d

=

7 and 1

=

188 but we do not known.

:. Using T,, =a

+

(n- l)d

188

=

-15

+

(n - 1)7

203

=

7n - 7

Hence n = 30

30 n

Now S30

=

2

(-15

+

188) using S,

=

l(a

+

l)

=

15 X 173

=

2595.

Example (iii):

+

188.

How many terms in the series 48

+

44

+

40

+ · · ·

need to be taken to give a sum of 308?

Solution:

n

Using S,

=

2

[2a

+

(n- l)d] where a= 48, d

=

-4 and S,

=

308

n

we have 308

=

2

[96

+

(n - 1) x ( -4)]

616

=

n[96 - 4n

+

4] 616 = lOOn - 4n2

:. n2 - 25n

+

154

=

0

:. (n - ll)(n - 14) = 0

:.n=

llor14.

Note that the sum of 11 terms = the sum of 14 terms = 308. What does this tell you about T₁₂

+

T13

+

T14?

In this method the quadratic equation sometimes yields solutions which are fractional or negative. Since n represents the number of terms, only positive integer solutions have meaning.

1. For the series :

(a) 5

+

12

+

19

+

26

+

(b) 6

+

8

+

10

+

12

+ .. .

(c) 20

+

17

+

14

+

11

+ .. .

(d) -17 - 8

+

1

+

10

+ .. .

(e) 1

t

+

3

+

4±

+

6

+ · · ·

find the sum of 10 terms. find the sum of 18 terms. find the sum of 14 terms. find the sum of 30 terms. find the sum of 24 terms.

2. Show that the sum of n terms of the series 1

+

2

+

3

+

4

+ · · ·

is given by S, = ~(1

+

n) and hence find the sum of all the integers from 1 to 100.

3. What is the sum of the first 30 positive even numbers?

4. Find the sum of 20 terms of an arithmetic sequence if the first term is 9 and the last term is 99.

5. Find the sum of all the positive multiples of 7 which are less than 100.

6. Find the sum of all the positive integers less than 100, which are not divisible by 6.

7. The first term of an arithmetic sequence is 3 and the twentieth term is 136. Find the common difference of the sequence and the sum of 20 terms.

8. Show that the sum of n terms of the series 1

+

3

+

5

+

7

+ · · ·

is given by S,

=

n2 and hence find the sum of the first 50 odd numbers 1

+

3

+

5

+

7

+ · · · +

99.

9. Find the sum of the series 2

+

3

+

4

+ · · · +

399.

10. The nth term of an arithmetic sequence is T,,

=

3n

+

1. Find the sum of 40 terms in this

sequence.

11. The nth term of an arithmetic series is 28 - 3n, find T_{1 ,} T₄₀ and S_{40 .}

12. How many terms must be taken in the series 5

+

11

+

17

+

23

+ · · ·

to make the sum 208?

13. How many terms in the series -3

+

0

+

3

+

6

+

9

+ · · ·

are needed to give a sum of 105?

14. Find the sum of all the integers between 100 and 200 which are multiples of 9.

15. A boy earns $10 the first week and 50 cents more each week than in the preceding week. What is the total sum of money he earns in 20 weeks?

16. A ball, rolling down a slope, rolls 16 em in the first second, 48 em in the second second, 80 em

in the third second and so on. At this rate how far will

it

roll in 8 seconds?

17. A drill test costs $200 for the first 10 metres, $250 for the next 10 metres, $300 for the next 10 metres and so on. What will the drilling cost for a depth of 180 metres?

18. A man saves $2000 in the first year of a savings programme and increases his yearly savings by $200 each year thereafter. How much has he saved at the end of 15 years?

19. The cost of building a multi-storey building is $300 000 for the first floor, $375 000 for the second floor, $450 000 for the third floor and so on. Find the total cost if the building is 8 storeys high.

20. The sum of 6 terms of an arithmetic series is 45, the sum of 12 terms is -18. Find the first term and the common difference.

21. Find the sum to 20 terms of the series whose nth term is 311 - 1.

22. How many terms of the series 23

+

19

+

15

+

11

+ · · ·

must be added to give a sum of 50?

23. The sum of n terms of a sequence is given by S, = 112

+

311. Find the first three terms of the

sequence and show that it is an arithmetic progression. What is the common difference?

24. The sum of 11 terms of a certain series is given by S,,

=

211

+

3112• Show the series is in arithmetic

progression and find:

(a) the common difference (b) the 20th term (c) the sum of20 terms.

25. The numbers x1 , x2 , x3 , x4 , • . . , x, are such that for 11 a positive integer x,

=

411

+

1. Show

the numbers form an arithmetic sequence and find the sum to 30 terms.

27. There are 10 apples in a row 5 metres apart. The first apple is 5 metres from a basket. How far does a boy run who starts at the basket and returns the apples to the basket, one by one?

28. A man moves a load of soil for top dressing an oval by emptying barrow-loads in a line 20 metres

apart, with the first heap 20 metres from the load of soil. How far does he walk if he empties

24 barrowfulls and returns to the load each time? ·

29. From a length of rod 600 em in length, 25 pieces are cut off, each 1 em longer than the preceding

piece. If the rod is exactly used up find the length of the first piece cut off.

Sigma notation is a shorthand method for indicating the sum of a series. The symbol

L

means sum.

6

L (

4n - 2) means sum the series for which T,, = 4n - 2 for values of n f