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Sequence & Series - Arithmetic
Sequences & Series
Sequences and Series: Arithmetic
What is the difference between a sequence and a series?
What is an arithmetic sequence?
What is the formula for the general term of an arithmetic sequences?
Suggest one practical application for the sum of an arithmetic series.
What is the difference between a sequence and a series?
What is an arithmetic sequence?
What is the formula for the general term of an arithmetic sequences?
Suggest one practical application for the sum of an arithmetic series.
Answer these questions, before working through the chapter.
Answer these questions, after working through the chapter.
But now I think:
What do I know now that I didn’t know before?
I used to think:
Sequences and series of numbers occur frequently in real life and mathematics. This booklet introduces the basic concepts focusing on arithmetic sequences and series. Several applications will also be discussed
Sequences and Series: Arithmetic
Sequences and Series: Arithmetic
Basics
A sequence is a list of numbers in a specific order like this 3, 5, 8, 9 Each number in a sequence is called a term of the sequence. Above, the 1st term is 3 and the 2nd term is 5. We write this as T
3
1= , T2= and so on. 5
Here are two examples.
Look at these sequences. a
a b
b
4, 7, 34, 0, 9, 7 -10, 14, 20, -22, 26
` The first 5 terms are 9, 13, 17, 21, 25. ` The first 5 terms are 4, -1, -6, -11, -16.
10 T1=- first term 14 T2= second term T3=20 third term T4=-22 fourth term
The number of terms of the sequence is 5.
T1= 4 first term
T2= 7 second term
T3=34 third term
T4= 0 fourth term
The number of terms of the sequence is 6.
T1 is the 1st term of the sequence, T2 is the 2nd term of the sequence. Tn is the nth term of the sequence, also called
the General term of the sequence. Here are two examples.
Write down the first 5 terms of these sequence with the formulas given.
The nth term of a sequence is given by the
formula Tn=4n+ . Use this with 5 n =1 2 3 4 5, , , , to write down the first 5 terms of this sequence.
The nth term of a sequence is given by the formula
Tn=9-5n. Use this with n =1 2 3 4 5, , , , to write
down the first 5 terms of this sequence.
4 5 4 5 4 5 4 5 4 5 T T T T T 1 9 2 13 3 17 4 21 5 25 1 2 3 4 5 = + = = + = = + = = + = = + = ^ ^ ^ ^ ^ h h h h h 9 5 9 5 9 5 9 5 T T T T T 9 5 1 4 2 1 3 6 4 11 5 16 1 2 3 4 5 = - = = - =-= - =-= - =-= - =-^ ^ ^ ^ ^ h h h h h
This sequence has 4 numbers.
The number of terms of the sequence is 4.
Sequences and Terms
Sequences and Series: Arithmetic
Basics
a b 12 5 7 ... T T T T 5 19 5 14 26 5 21 5 7 5 7 5 7 5 7 0 1 2 3 1 2 3 4 # # # # ` = = = + = = + = = + = + = + = + = + ... T T T T 125 115 125 10 105 125 20 95 125 30 125 10 125 10 125 10 125 10 0 1 2 3 1 2 3 4 # # # # ` = = = -= = -= = -= -= -= -= -T T n T n n 5 7 5 7 7 7 2 1 n n n # ` = + = + -= -^ h T T n T n n 125 10 125 10 10 135 10 1 n n n # ` = -= - + = -^ hHere are two examples where you need to find the formula for the general term of a sequence.
Find the formula for the general term Tn for the sequences.
The nth term of a sequence is given by T
n
9 8
n= + . For which value of n does the nth term equal 161?
5, 12, 19, 26, 33, 40, ... 125, 115, 105, 95, 85, ... The terms are increasing by 7. The terms are decreasing by 10.
Here is an example where a term is given and the n value found.
Substituting into the formula gives
n n n n n 161 9 8 9 161 8 9 153 9 153 17 ` = + = -= = =
Sequences and Series: Arithmetic
Questions
Basics
For the sequences shown, fill in the values indicated 1. a a c a a a b b b b d b 3, 6, 1, -7, 0, 45 9, 6, 2, -3, -10, 20, 4
The number of terms in the sequence = The number of terms in the sequence =
T T T T 1 2 3 4 = = = = T T T T 1 2 3 4 = = = =
Write down the first 5 terms of the sequence whose general term is given: 2. 3. 5. 4. Tn=7n Tn=1-3n Tn=100+8n Tn=-25-6n Tn=6n+7 19 Tn=- n
What is the 5th term of the sequence with nth term:
The nth term of a sequence is given by
Tn=8n- . For which value of n does 5 the nth term equal 163?
The nth term of a sequence is given by
Tn=3-12n. For which value of n does
the nth term equal -177?
Try and find the general term, Tn, of these sequence
Sequences and Series: Arithmetic
Basics
T T T T 10 10 5 15 15 5 20 20 5 25 1 2 3 4 = = + = = + = = + = T T d 2 6 8 2 1 ` = + = + = T T d 20 6 26 5 4 ` = + = + =An Arithmetic Sequence is a specific type of sequence where the difference (d) between consecutive terms is constant (i.e. the same)
Examples of arithmetic sequences
A sequence has T1=10 and a common difference of 5. Find T2, T3, T4.
If the following sequence is arithmetic find the missing terms: 2, T2, 14, 20, T5, 32, 38
a An arithmetic sequence with 5 terms b An arithmetic sequence with 6 terms
4, 8, 12, 16, 20 32, 27, 22, 17, 12, 7 , 4 T1= , 8 T2= and so on. , T1=32 , 27 T2= and so on. or . d d T T T T 8 4 12 8 4 4 2 1 3 2 = -= -= -= -= = or 5 d d T T T T 27 32 22 27 5 2 1 3 2 = -= -= -= -
=-The common difference is 4. The common difference is -5.
+4 +4 -5
d d d
-5
+4 +4 -5 -5 -5
To find the common difference, find the difference between consecutive terms.
Here is an example where the first term and common difference are used to find other terms.
Start with 10 and increase by 5 each term.
So the arithmetic sequence is: 10, 15, 20, 25.
The common difference can be used to find missing terms.
First, find d, by subtracting tow consecutive terms. d=20-14=6 Use T1 and d to find T2
Use T4 and d to find T5
... ... d=T2-T1=T3-T2= =Tn-Tn-1= ... , , , , T1 T2 T3 T4 `
The difference, d, is called the common difference, and in general we say: d=Tn-Tn-1. An arithmetic sequence
can also be called an Arithmetic Progression and the abbreviation AP is commonly used.
Arithmetic Sequences
Sequences and Series: Arithmetic
Questions
Basics
For each of these arithmetic sequences below, find 6. 7. 8. 9. 10. 11. (i) (ii) (iii) (iv)
the number of terms in the sequence the first term, T1
the fifth term, T5 the common difference, d
a c e a a a b b b b d f 12, 16, 20, 24, 28, 32, ... 1, 7, 13, 19, 25 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 3, 8, ... 17, ... 5, T2, 19 ... -6, T2, 8 ... -19, -14, ... 3, -1, ... -10, -7, -4, -1, 2, 5, 8 -5, 7, 19, 31, 43, 55 4.2, 5.8, 7.4, 9.0, 10.6, 12.2
What are the next three terms if the common difference for these sequences is 5?
What are the next three terms if the common difference for these sequences is -4?
What is the second term if the common difference for these sequences is 7?
The fifth and seventh terms of an arithmetic sequence are 19 and 27 respectively. Find d.
Sequences and Series: Arithmetic
Basics
The letter ‘a’ is used to represent the first term of a sequence. The letter ‘d’ is used to represent the common difference of an arithmetic sequence. For example
8, 18, 28, 38, ...
2, 5, 8, 11, 14, 17, ...
The common difference is d=18-8=10 The first term is a=8
The next step is to work out a formula for any term of an arithmetic sequence. Here is an example.
Finding the general term (also called the nth term) of an arithmetic sequence
The sequence 3, 10, 17, 24, 31, 38, 45, 52 has a common difference of d = 7
T T T T 3 10 3 7 17 3 7 7 3 2 7 24 3 7 7 7 3 3 7 1 2 3 4 = = = + = = + + = + = = + + + = + ^ ^ h h
Can you see the pattern?
The general term is given by Tn=3+7^n-1h, and this simplifies to give Tn=7n-4
Starting from ‘a’ and adding the common difference ‘d’ for each term, we get:
The formula for each term of an arithmetic sequence a, a + d, a + 2d, a + 3d, ... is:
T1=a T T d a d a 2 1 d 2= 1+ = + = +^ - h 2 3 T a n d n n T n 1 1 2 3 3 3 1 n n # = + -= + -= + -= -^ ^ h h T T d a d a d 2 3 1 3= 2+ = + = +^ - h T T d a d a d 3 4 1 4= 3+ = + = +^ - h T T d a d a d 4 5 1 5= 4+ = + = +^ - h
T
n=
a
+
^
n
-
1
h
d
n = number of the term in the sequence nth term
d = common difference a = first term
This is the General Term of an Arithmetic Sequence.
Finding the general term of the following arithmetic sequence using the formula.
The arithmetic sequence
has first term a = , and the common difference is d2 =5-2= , so the formula for the n3 th term is
The 10th term is given by T
3 10 1 29
10= ^ h- = .
The 15th term is given by T
3 15 1 44
15= ^ h- = .
Sequences and Series: Arithmetic
Basics
Here is an example of using the formula to find ‘d’.
The common difference of an arithmetic sequence is 14 and the twelfth term is 179. Find the first term.
The common difference of an arithmetic sequence is -8 and the twelfth term is -144. Find the first term. Find d if a= and T5 7=71
This is given: a=5 n=7 T7=71
Substituting these into the formula Tn=a+^n-1hd gives:
d d d 71 5 7 1 71 5 6 6 71 5 11 ` ` = + -= + = - = ^ h
` The common difference is d=11
Sometimes a term that isn’t the first term is used to find terms of a sequence. In the next example the general term is needed to find the first term, T1= .a
The next example has a negative d. This is given: This is given: 14 179 d= T12= 8 16 144 d=- n= T16
=-Substituting these into the formula Tn=a+^n-1hd gives:
Substituting these into the formula Tn=a+^n-1hd gives:
179 14 179 14 a a a 12 1 11 25 = + -= -= ^ ^ h h a a a 144 16 1 8 144 15 8 6 - = + - -=- + = ^ ^ ^ ^ h h h h
So, the first term is a=25.
Sequences and Series: Arithmetic
Basics
Here is an example of finding the nth term of a sequence.
For the arithmetic sequence -12, -7, -2, 3, 8, 13, 18, ...
In a sequence T8=149 and T12=225. Find the general term.
a
c
b
d Find a
Find an expression for the nth term T
n. Find d Find T20 a=T1=-12 12 5 T a n d n T n 1 1 17 5 n n ` = + -=- + -=- + ^ ^ h h d=T5-T4=8-3=5
In the next example, two non-consecutive terms are used to find the general term of the sequence, Tn.
The general term is used to form a system of two simultaneous equations which can be solved.
T 17 5 20
83
20=- +
=
^ h
By substituting the information into the formula Tn=a+^n-1hd two equations are formed, and these need
to be solved simultaneously for a and d. Substituting gives us
Solve these equations:
Subtracting 2 - 1 gives:
Substituting into 1 gives:
149 a 8 1 d ` = +^ - h a d 149= +7 a d 225= +11 225 149 11 7 76 4 d d d d 19 ` - = -= = 225 a 12 1 d ` = +^ - h and 1 2 149 7 a a 19 16 ` = -= ^ h 16 19 T n T n 1 19 3 n n ` ` = + -= -^ h Tn = 149 , n = 8 T12 = 225 , n = 12
Sequences and Series: Arithmetic
Questions
Basics
Find the formula for the nth term, T
n, of an arithmetic sequence with first term, a = 9, and common
difference, d = 12. 12. 13. 14. 15. 16. 17.
For the arithmetic sequence 21, 16, 11, 6, 1, -4, -9:
The fifth term of an arithmetic sequence is T5=17. Using the formula Tn=a+^n-1hd, and given that
the first term is a= , find the common difference d.1
The first term of an arithmetic sequence is 21 and the tenth term equals -78. Find d.
A sequence has a common difference 9 and T20=174. Find the first term.
An arithmetic sequence has T5=45 and T8=27. Find the general term by substituting into
Tn=a+^n-1hd and solve a pair of simultaneous equations for a, d.
a
c
b
d Find a
Find an expression for the nth term T
n.
Find d
Sequences and Series: Arithmetic
Knowing More
A series is the sum of the terms in a sequence.
Comparing a sequence to a series.
Consider the series 2+4+6+8+10+12+14+16+...
5, 2, 13, 10, 8 is a sequence. 5 + 2 + 13 + 10 + 8 is a series.
Tn is used as a symbol for the nth term. Terms in a series are found the same way as terms in a sequence.
Sn is the notation for the sum of the first n terms. Here are some basic examples.
The second term of the sequence is T2= .4
The sum of the first two terms of the sequence is written b
a
The fifth term is T5=10.
The sum of the first five terms is written
S2=T1+T2=2+4=6
S5=T1+T2+T3+T4+T5=2+4+ + +6 8 10=30
A series has 16 terms and is given that S15=540 and S16=600 . Find the value of the 16th term, T16 .
The next example highlights the definition of the sum to n terms.
By definition, S15=T1+T2+...+T15 S16=T1+T2+...+T15+T16 Subtracting these two leaves us with just T16.
Therefore,
... ...
S16-S15=^T1+T2+ +T15+T16h-^T1+T2+ +T15h=T16
T16=600-540=60
Series
Given a series with 16 terms: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53. Find S1, S4, S10, S16 2 S1=T1= S T T T T 2 3 5 7 17 4= 1+ 2+ 3+ 4 = + + + =
To find S16 the sum S10 can be used.
2 3 5 7 11 13 17 19 23 29 S T T T T T T T T T T 139 10= 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 9+ 10 = + + + + + + + + + = S S T T T T T T 129 31 37 41 43 47 53 381 16= 10+ 11+ 12+ 13+ 14+ 15+ 16 = + + + + + + = This example finds the sum to n terms for serveral n values.
Sequences and Series: Arithmetic
Knowing More
Remember that in an arithmetic sequence, consecutive terms have the same difference. This holds in an
arithmetic series also, consecutive terms always differ by the same amount, d, the common difference. The terms
Tn + 1 and Tn are consecutive and for an arithmetic series:
Arithmetic Series
Find the missing terms if the given series are arithmetic.
Find the missing terms of the series 20+T2+T3+116 if it is an arithmetic series. a a b b T 13+30+ 3 d T T 30 13 17 2 1 = -= -= T 30 d 30 17 47 3= + = + = 9+T2+31 and 31 9 T3= a= 9 d 31 d 3 1 9 2 31 ` ` + - = + = ^ h Solve for d, d T 2 31 9 11 9 11 20 2 = - = = + = So
Here is an example of an arithmetic series with two unknown terms. Here are some arithmetic series.
Consecutive terms have a difference of d.
and 116 20 T4= a= 20 3 116 d d d d 20 4 1 116 3 116 20 32 ` + - = + = = -= ^ h
So the missing terms are T2=20+32=52
Examples of arithmetic series.
S7=1+ +9 17+25+33+41+49 S3=10+ + +8 6 4
The common difference is d = 9 - 1 = 8. The common difference is d = 8 - 10 = -2. Here are some basic examples to help understand arithmetic series
Sequences and Series: Arithmetic
Knowing More
Find the sum of the first 25 terms of the arithmetic series 4+10+16+22+28+34+...
The third term of an arithmetic series is given by T3=56 and the sum of the 3rd and 4th terms is T3+T4=140.
a b
What about the general sum, Sn? It is important to find a formula for Sn based on a, n, and d. Here is the derivation.
The sum of the first n terms of an arithmetic series with first term, a, and common difference, d, is given by
S a S a n 1 d n n = =^ +^ - h h ... ... ... S S a a n d a d a n d a n d a d a n d d S a n d a n d a n d a n d n n 1 2 2 1 2 2 1 2 1 2 1 2 1 1 2 1 n n n ` ` + = + + - + + + + - + + + - + + + + - + = + - + + - + + + - + + -^ ^ ^ ^ ^ ^ ^ ^ ^ ^ h h h h h h h h h h 6 6 6 6 6 6 6 6 @ @ @ @ @ @ @ @ a d a n 2 d + + + + -^ ^ ^ h h h ... ... a n d a d 2 + + + -+ + + ^ ^ ^ h h h a n d a 1 + + -+ ^ ^ h h In reverse:
Adding these two rows gives:
S n a n d
2 2 1
n= 6 +^ - h @
The common difference d The first term a
This is important if you want to find S100 or S200. Here is an example.
, 4 a=T1= d=10-4=6 n=25 2 2 6 S n a n d 2 1 2 25 4 25 1 1900 25= + -= + -= ^ ^ ^ h h h 6 6 @ @
The next example uses the sum of two terms to find a and d, which are used in the sum formula.
Find the fourth term, T4
56 T T T 140 3 3 4 = + = Subtracting gives T4=140-56=84
Find the sum to 40 terms, S40.
d T T a T T d T d 84 56 28 2 4 3 1 2 3 = - = - = = = - = -a 56 2 28 0 ` = - ^ h= 2 2 28 S n a n d S 2 1 2 40 0 40 1 21840 40 40 = + -= + -= ^ ^ ^ h h h 6 6 @ @ The sum to 25 terms is 1900.
The number of terms, n
Sum of an Arithmetic Series
There are n lots of a^2 +^n-1h hd
S n a n d
2 n 2 1
` = 6 +^ - h @
Finally dividing by 2 gives the general term.
Sequences and Series: Arithmetic
Knowing More
Find the sum of the arithmetic series 5+10+15+20+25+...+2555.
An arithmetic series has sum S n n
6
5 1
n= ^ + h. How many terms must be taken for the sum to exceed 200?
The sum of the first n even numbers is an arithmetic series and Sn=2+4+6+8+...+2n
a=5 d =5 l=2555
Step 1: Find n. Step 2: Substitute known values
T a n d n n 1 5 1 5 5 n= + -= + -= ^ ^ hh T n n 2555 5 2555 5 2555 511 n ` = = = = S n a l 2 2 511 5 2555 654080 511 ` = + = + = ^ ^ h h
If the last term of the series l is known then the formula S n 2a n d
2 1
n= 6 +^ - h @ becomes
S n a l
2
n= ^ + h
The last term is l=a+^n-1hd
The first term a
Sometimes n needs to be found before being able to calculate Sn. Use the formula for the last term Tn to find the
number of terms, n.
a What is the general term, Tn? b Use the formula S n a l
2
n= ^ + h to find the
formula for the sum to n terms, Sn.
The last term in the series sum gives away the general term, T` n=2n.
Check that the formula gives 2, 4, 6, 8 … when substituting = 1, 2, 3, 4 … .
a = 2, l = 2n and since there are n terms
the sum is 2 2 S n a l n n n n n n 2 2 2 2 1 1 n= + = + = + = + ^ ^ ^ ^ h h h h
Use trial and error for different values of n.
When n = 5, S 25 6 5 5 6 5= ^ ^h h= . When n = 10, S 6 5 10 11 3 275 n= ^ ^h h= . When n = 15, S 200 6 5 15 16
15= ^ ^h h= . This is still not greater than 200 so we increase n by 1 again..
When n = 16, S = 5 16 17^ ^h h= 680 2200
Sequences and Series: Arithmetic
Questions
Knowing More
a a a b c b c bIf these are arithmetic sequences find the value of the missing terms 18.
Find the sum to 25 terms of the arithmetic series using the formula S n a n d
2 2 1
n= 6 +^ - h @.
Find the sum of the series using the first and last term formula, S n a l 2 n= ^ + h. T 6+25+ 3 ... 32+30+28+26+24+22+20+18+16+14+12+ ... 25 21 17 13 9 5 1 3 7 - - - + + + ... 5+ + - - -3 1 1 3 5- - -7 9 11-13-15-17 -T 20+ 2+4 c 12+T2+T3+39 3+ +7 11+15+19+23+27+31+35+39+43 9 16 23 30 37 44 51 58 65 72 - - - -1 10 19 28 37 45 54 63 72 81 90 99 - - - -19. 20.
Sequences and Series: Arithmetic
Questions
Knowing More
a
c d
b Find the sum of these series.
An arithmetic series has 20 terms with first and last terms 5 and 195 respectively. Find the sum of the series.
... 2+4+ + +6 8 10+ +200. ... 1 0 1 2 3 4 199 - + + + + + + + 250+238+...-134 ... 93+87+81+ + +6 3 21. 22.
Sequences and Series: Arithmetic
Questions
Knowing More
a
b
c
A series has T3= and 4 T2+T3=-12. Find T2.
The fourth term of an arithmetic series is 28 and the sum of the fourth and fifth terms is 32.
Find the fifth term.
Find the common difference, d.
Using the formula for the nth term, T
a n 1 d
n= +^ - h with n = 5 and your answers from above, to find
the value of a.
a
b
c
For the arithmetic series 1744+...+28+22+16+10+4
Find the values of a and d and write down the nth term T
n.
By using the last term, 4, and the formula for Tn , find out how many terms are there in the series?
Find the sum of this series.
23.
24.
Sequences and Series: Arithmetic
Questions
Knowing More
a b c d e f gFind the sum of the first 5 multiples of 3, that is, 3+ + +6 9 12+15.
Is this an arithmetic sequence? If so, find d.
The sum of the first n multiples of 3 is an arithmetic series and Sn=3+ + +6 9 12+...+3n
Use the formula S n a l 2
n= ^ + h to find the sum to n terms and show this sum is S n n
2
3 1
n= ^ + h
Find the general term Tn. 26.
Find n if the sum is 315.
How many terms must be taken for the sum to exceed 560? (Use trial and error approach)
Sequences and Series: Arithmetic
Using Our Knowledge
The starting salary of an accountant is $34 000 per year. After each year of employment he will receive an increase in the salary by an amount $1200 .
a
a
What is his salary in the ninth year of employment? Salary for first year = $34 000
Salary for second year is: $34 000 + $1200
Salary for third year is: $34 000 + $1200 + $1200 $34000 $1200#2
= +
` Salary for ninth year is $34000+$1200#8
Or using the formula for arithmetic series with: $34000 a = d =$1200 n=9 $ $ T a n d T n 1 34000 1200 1 n n = + -= + -^ ^ h h Therefore, $34000 $1200 $43600 T9= + ^9-1h=
How much are his total earnings for the first nine years.
The total salary in the first nine years is the sum to 9 terms of an arithmetic series. Denote the total earnings to the end of the nth year by S
n (sum to n terms). Since we know n= and the last term is 9
l=43600 then the formula for the sum to n terms of an arithmetic series gives:
S n a l 2 2 9 34000 43600 349200 n= + = + = ^ ^ h h
Total earnings are $349 200
This is the last term in the series
Arithmetic series have uses in the real world. The next example is an application of arithmetic series to a financial problem involving annual salary increases.
Sequences and Series: Arithmetic
Using Our Knowledge
Here is another real world problem.
A section of a stadium has 7 seats in the first row, 10 seats in the second row, 13 seats in the third row, and so on. If the last row has 91 seats, how many seats are in this section?
This can be expressed as an arithmetic sum as:
... a d 7 10 13 91 7 3 + + + + = =
To find the sum, the number of terms, n, is needed. The nth term of this series is given by:
7 3 T a n d n n 1 1 3 4 n= + -= + -= + ^ ^ h h
The last term is 91. Which term is this? Solve:
91 3 4 91 3 87 T n n n 29 n= + = = =
So there are 29 terms. The sum of the arithmetic series using the formula is:
S n a l S 2 2 29 7 91 1421 29 29 = + = + = ^ ^ h h
First row, 7 seats
Second row, 10 seats
Third row, 13 seats
Last row, 91 seats
. . .
.
.
Sequences and Series: Arithmetic
Using Our Knowledge
... ... 2 1 20 2 2 20 2 3 20 2 32 20 40 80 120 160 1280 # # + # # + # # + + # # = + + + + +This is an arithmetic series.
The sum of the arithmetic series is:
(last term) 40 40 32 1280 a d n l = = = = S n a l 2 2 32 40 1280 21120 32= + = + = ^ ^ h h 1 2 32 40 m 640 m 20 m Emptying wheelbarrow
Going back to refill the wheelbarrow
Here is another more complicated real world problem.
A groundskeeper distributes fertilizer on the green using a wheelbarrow. Over several days he empties the wheelbarrow in 20 metre increments, and has to go back to refill each time. If he empties 32 barrowfulls, how far does he walk?
For the first load he walks 20 m and returns, so he has travelled 2# #1 20m. For the second load he walks m
2#20 and returns, so he has travelled 2# #2 20m and so on. In total he will travel
Sequences and Series: Arithmetic
Using Our Knowledge
Here is an example relating to building.
a
b
Find the number of boards.
45 S n a l n n n 2 270 2 4 8 12 2 270 n= + = + = = ^ ^ ^ h h h a = 4, l = 8
The difference in length between adjacent boards is the common difference d.
The last board is number 45.
Tn=a+^n-1hd m cm ( d.p.) 4 8 . . d d 45 1 11 1 0 091 9 09 2 ` + - = = = = ^ h 4 m Floor 8 m
A floor in room has the shape of a trapezium as shown. It is to have floor boards put in. The difference between the lengths of adjacent boards is a constant and so the lengths of the boards form an arithmetic sequence. The shortest board is 4 m in length and the longest board is 8 m. The sum of the lengths of the boards is 270 m.
Sequences and Series: Arithmetic
Questions
Using Our Knowledge
The annual salary of a sales manager increases by $750 each year, where in his first year he earned $45 000. a
b
What is his salary in the sixth year of employment.
How much money did he earn in total in the first 5 years of employment?
27.
28. Chairs in an amphitheatre are arranged in an increasing order where the number of chairs in a row is increasing from 6 until 58. Find the number of chairs if the difference in the number of chairs in rows next to each other is 2.
Sequences and Series: Arithmetic
Questions
Using Our Knowledge
An architect is finding the cost of building a multi storey building. The 1st floor costs $200 000, the 2nd floor
costs $225 000 and each subsequent floor costs $25 000 more to build than the floor below. What does it cost to build a 40-storey building?
The temperature in a high pressured hot water tank is falling at a constant rate. A reading was taken each 10 minutes and these were:
The final reading taken was equal to 26c. How many readings were taken altogether.
, , , ...
230 224 218c c c 29.
Sequences and Series: Arithmetic
Thinking More
The Greek sigma (/ ) symbol is used to write series easily. / is a short hand notation for describing series in a precise and concise way. Writing a sum in sigma notation relies on noticing the pattern in the series. In sigma notation, we have
T
nT
T
T
T
T
n 1 2 3 4 5 1 5=
+
+
+
+
=/
The general expression for the nth term is T
n
Formula
Start
n = 1 n = 2 n = 3 n = 6 n = 8
End
Write out the terms of the series given in sigma notation as 2n
n 1 8 =
/
.Write out the series in expanded form 4n 1
6
n 0
+
=
/
.The bottom number, ‘n= ’, tells you to start the first term by substituting in n1 = . 1
Then substitute n= and add it on, then n2 = and add it on again, all the way up to n3 = . 8 Expanding the sigma notation one gets
2
n
2
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
4
6
8
10
12
14
16
n 1 8=
+
+
+
+
+
+
+
=
+
+ + +
+
+
+
=^
h
^
h
^
h
^
h
^
h
^
h
^
h
^
h
/
Here is another example.
n 4 1 4 0 1 4 1 1 4 2 1 4 3 1 4 4 1 4 5 1 4 6 1 1 5 9 13 17 21 25 n 0 6 + = + + + + + + + + + + + + + = + + + + + + = ^ ^ h h ^ ^ h h ^ ^ h h ^ ^ h h ^ ^ h h ^ ^ h h ^ ^ h h
/
Sigma Notation
n 2 n 1 8 =/
The last value of n is n = 5
Sequences and Series: Arithmetic
Thinking More
Write out the series in expanded form 4 n
n 3 2
-
=-/
.Write out in expanded form (without sigma notation).
Write the series in sigma notation: 5 11 17 23 29 35 41 47+ + + + + + +
n 4 4 3 4 2 4 1 4 0 4 1 4 2 4 3 4 2 4 1 4 3 2 7 6 5 4 3 2 n 3 2 - = - - + - - + - - + - + - + -= + + + + + + + + = + + + + + =-^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ hh hh hh h h h h h h
/
Here is an example with negative numbers where more care is needed.
Here a two examples of writing the expanded form from sigma notation.
a b n 5 4 n 2 6 + =
/
n 5 7 n 3 5 - -=/
14 19 24 29 34 5 2 4 5 3 4 5 4 4 5 5 4 5 6 4 120 = + + + + + + + + + = + + + + = ^ ^ h h ^ ^ h h ^ ^ h h ^ ^ h h ^ ^ h h 5 7 5 7 15 7 20 7 25 7 3 4 5 5 7 81 = - - + - - + - -=- - - - =-^ h ^ h ^ h 6 @ 6 @ 6 @Some examples of writing a series in the sigma notation follow.
This is an arithmetic series with common difference, d=11-5= , and a6 = .5
Therefore the sum is
Tn 5 6 n 1 6n 1 ` = + ^ - h= -n 6 1 n 1 8 -=
/
Start at n = 2 and end at n = 6
Sequences and Series: Arithmetic
Thinking More
Calculate the sum of the series: 7m 4
m 0 100
+
=
/
Write it out in expanded format
... 7m 4 4 11 18 704 m 0 100 + = + + + + =
/
This is an arithmetic series,, , , 4 7 704 101 a= d= l= n= S n a d n S 2 2 1 2 101 2 4 7 101 1 35754 n 101 # = + -= + -= ^ ^ ^ ^ hh hh (number of terms) The next example involves recognising that the series is arithmetic and finding the sum.
Sequences and Series: Arithmetic
Questions
Thinking More
Write out in expanded form (without sigma notation) and evaluate
Write these series in sigma notation
Calculate the sum of each of these series. a a a b b b j 7 j 0 5 =
/
6+12+18+24+30+36 j 9 5 j 1 7 + =/
4 6n n 90 233 -=/
k 4 9 k 1 8 -=/
2+ +9 16+23+30 31. 32. 33.Sequences and Series: Arithmetic
More Questions
Basics
1.2.
3.
What is the general term of the sequence 6, 8, 10, 12, ...?
What is the 3rd term of the sequence with general term T
n 4 3 n= + ? a a a a b b b b c c c c d d d d Tn=2+4n Tn=4n Tn=11-7n 2 Tn=4+ n Tn=3+5n , , , , ... 7 1 7 2 7 3 7 4 Tn=6n+2 Tn=9-n Tn=-3 4^ -5nh Tn=6+8n 7 14 Tn=- n
-If the numbers 12, x, 24 are in arithmetic progression, what is the value of x? 7 15 12 16 15 18 67 20
4. If Tn=19-9n, what is the value of T3?
-8 -7 7 8 5. a a a a a b b b b b c c b b b a a a d Which of the following does not represent an arithmetic series?
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
For the sequences shown, write down the number of terms, and the values of T1, T3, T6.
Write down the first 5 terms of the sequence whose general term is given
What is the 5th term of the sequence with nth term
The nth term of a sequence is given by T
n
8 5
n= - . For which value of n does the nth term equal -162?
The nth term of a sequence is given by T
n
11 29
n= + . For which value of n does the nth term equal 260?
3, 6, 1, -7, 0, 45 2, 22, 42, ... 9, 6, 2, -3, -10, 20, 4 -6, -1, 4, 9, 14, 19, ... -16, -8, T3, ... 5, T2, 21, ... -19, ... -19, ... 7, ... 34, ... Find the general term, Tn , of these sequence
What are the next three terms if the common difference is 7? What are the next three terms if the common difference is -8? What is the missing term if the common difference is 8?
The fourth and eleventh terms of an arithmetic sequence are 69 and 167. Find the common difference. The first term of an arithmetic sequence is 19 and the common difference is -9. What is the seventh term of the sequence?
Find the formula for the nth term, T
n, of an arithmetic sequence whose first term, a, and common
difference, d, are given by a = 13 and d = -8.
Sequences and Series: Arithmetic
More Questions
a a b b c c 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. Find a, d 20, x, 30 13, 20 Find T20For the arithmetic sequence 12, 25, 38, 51, ...
The fifteenth term of an arithmetic sequence is T15=20. Using the formula Tn=a+^n-1hd, and given
that the first term is a = 188, find the common difference d.
The first term of an arithmetic sequence is -26 and the eleventh term equals 64. Find the common difference. The common difference of an arithmetic sequence is 17 and the twentieth term equals 347. Find the first term. The sixth and tenth terms of an arithmetic sequence are given by T6=- and T7 10=-27. Find the general
term by substituting into Tn=a+^n-1hd, and solve a pair of simultaneous equations for a, d.
Is 35 a term of the sequence with general term Tn=76-4n?
Which term of the sequence 5, 13, 21, 29, 37, 45, ... is 341?
Write down the numbers 18 and 46 and insert three numbers between them so as to give 5 numbers in arithmetic progression.
The sum of the first two terms of an arithmetic sequence is 10 and the sum of the next two terms is 18. Find the first term and common difference.
If the sequence , ,
a b c
1 1 1 is arithmetic, show that ac b a c2 = ^ + h. Find x and the common difference if the following are AP’s
Find the common difference of the AP, then find x, given that T10=25;
If the sequence , , ,8 T T2 3 29 is arithmetic, what are the missing terms?
What is the sum of the series - - -4 3 ...+42?
How many terms are there in the arithmetic series 5+ + +7 9 11+...+31? Find the expression for the nth term, T
n. x + 2, 10, x + 8 13.25, 21.125 5x - 2, x + 5, 3x 15, 22 16, 24 7x - 12, 7x - 2, 7x + 8
Knowing More
S 2 47 4 42 47= ^- + h S 2 46 4 42 46= ^- + h S 2 45 4 42 45= ^- + h S 2 46 4 42 47=- ^- + h 32. 33. 13 -84The third term of an arithmetic series is given by T3=28 and the sum of T3+T4=-56 is given.
What is the fourth term T4?
If the sums S9=65 and S8=92, what is T9?
14 -28 19 28 20 84 a b c d a b c d a b c d a b c d
Sequences and Series: Arithmetic
More Questions
a a a a a a a b b b b b b d e f g c c d c c c c b c 34. 35. 36. 37. 38. 39. 40. 41. 42.If these are arithmetic series find the value of the missing terms
Find the sum to 25 terms of the arithmetic series using the formula S n a n d
2 2 1
n= ^ +^ - h h.
Find the sum of the series using the first and last term formula, S n a l 2
n= ^ + h
Find the sum of these series by first finding the number of terms, n, in the series using the last term and the formula Tn=a+^n-1hd
An arithmetic series has 41 terms with first and last terms -100 and 140 respectively. Find the sum of the series. The seventh term of an arithmetic series is given by T7=29 and the sum of T7+T8=-905 is given.
What is the eight term T8?
The fifth term of an arithmetic series is -1 and the sum of the fourth and fifth terms is -15.
For the arithmetic series 13 + 17 + 21 + ... + 6613.
T
23+50+ 3 -25+T2+1 6+T2+T3-9
62 + 60 + 58 + 56 + 54 + 52 + 50 + ...
1 + 2 + 3 + 4 + ... + 1000
Find the values of a and d and write down the nth term T
n.
Find the sum of the first 9 multiples of 6, that is, 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54. -90 - 81 - 72 - 63 + ...
-4 - 9 - 14 - 19 - 24 - 29 - 34 - 39 - 44 - 49 - 54 - 59 - 64 - 69 - 74 - 79
By using the last term, 6613, and the formula for Tn, find out how many terms are there in the series?
Is this an arithmetic sequence? If so, find d.
Use the formula S n a l 2
n= ^ + h to find the sum to n terms and show this sum is Sn=3n n^ +1h
Find n if the sum is 2610.
How many terms must be taken for the sum to exceed 7650? (Use trial and error approach, or logs) Give a simple reason why can this sum never be 695?
3 + 6 + 9 + 12 + ... + 1098
Find the fourth term.
-4 - 8 - 12 - 16 - 20 - ... - 516
Find the common difference, d. -23 - 16 - 9 - 2 + ... + 110
Using the formula for the nth term, T
a n 1 d
n= +^ - h and your answers from above, to find the value of a.
-1022 - 1000 - ... - 142 -1 - 2 - 3 - 4 -5 - 6 - 7 + ...
50 + 29 + 8 - 13 - 34 - 55 - 76 - 97 - 118 - 139 - 160 - 181 - 202 - 223 - 244 - 265
Find the sum of this series.
The sum of the first n multiples of 6 is an arithmetic series and Sn = 6 + 12 + 18 + 24 + ... + 6n.
Sequences and Series: Arithmetic
More Questions
a b a b 43. 49. 50. 51. 52. 53. 54. 55. 56. 57. 45. 46. 47. 48. 44. 37 4, 16, 64 $38 750 3, 27 2 + 3 + ... + 40What is his salary in the fourth year of employment?
How much money did he earn in total in the first 7 years of employment?
4 + 6 + ... + 44 -5 - 4 - ... + 26 2 + 4 + ... + 40 138 32, 64, 128 $40 000 1, 24 147 64, 128, 192 $41 250 1, 34 Find the sum of all positive multiples of 9 which are less than 230.
Three numbers x, y, z are inserted between 0 and 256, so as to give 5 numbers in AP. What are the values of
x, y, z?
A man earns $30 000 in his first year at work, and gets an increase in his salary by $1250 each year. What is his salary in the 8th year of his employment?
The sequence x, 7, 17, y are in arithmetic progression. What are the numbers x, y?
Which of the following is an arithmetic series with 21 terms whose sum is 504?
The annual salary of a tradesman increases by $1250 each year, where in his first year he earned $32 000.
An architect is finding the cost of building a multi storey car park. The first floor costs $170 000, the second floor costs $190 000, and $60 000 for each additional floor. What is the cost of building a 12 storey car park? A new building has 26 floors. The cost of building each floor varies. The first floor costs $2 000 000. The cost of building each subsequent floor will be $650 000 more than the floor immediately below.
Find the sum of the integers between between 150 and 250 which are multiples of 7.
The sum of the first 5 terms of an arithmetic series is -10, and the sum of the next two terms is -32. Find a, d. Find the sum of the AP: x+2x+3x+...+nx.
Find the sum: 18+ 50+ 98+...+45 2
Show that the sum to n terms of the AP 1 + 3 + 5 + ... is Sn=n2
Find the sum of the first 100 odd numbers.
114
16, 64, 128
$39 250
-3, 27
Using Our Knowledge
a b c d a a b b c c d d a b c d a b c d
The Chairs in a small amphitheatre are such that the first row has 9 chairs, and each row increases by 4 until the last row, which has 33 chairs. What is the total number of chairs.
Chairs in an amphitheatre are such that the first row has 12 chairs, the second row has 16 chairs, and each row increases by 4, until the last row, which has 104 chairs. Find the total number of chairs.
Sequences and Series: Arithmetic
More Questions
a a b b c 58. 59. 60. 61. 62. 63. 64. 65. 66. 67.How far does Paula swim in the fourth week?
Find the number of planks.
In which week does she first swim 4 km?
Find the difference in length between adjacent planks. What is the total distance Paula swims in 30 weeks?
Paula is training for a 4 km swimming race by swimming each week for 30 weeks. She swims 200 m in the first week, and each week after that she swims 200 m more than the previous week, until she reaches 4 km in a week. She then continues to swim 4 km each week.
The temperature in a cool room was taken at regular intervals after it was turned on, and the readings in degrees Celsius were 25 24.1 23.2 ...c, c, c, . Assume that these readings are in arithmetic progression. If the final reading taken was equal to -9 2c. , how many readings were taken altogether?
What is the expanded form of the series 9m
m 1 6
=
/
?What is the expanded form of the series 3k 3
k 2 0 3 -=
/
?What is the sigma notation for the series 4 + 7 + 10 + 13 + 16 + 19?
What is the sum of the series 3k 3
k 3 12
+
=
/
?Write out in expanded form (without sigma notation) and evaluate Write these series in sigma notation.
Calculate the sum of each of these series.
Thinking More
a c b a a b b 9 + 18 + ... + 54 3 - 1 + 9 + 24 1 + 9 + ... + 54 -3 + 0 + 9 + 24 9 + 18 + ... + 56 -3 + 0 + 3 + 6 1 + 18 + ... + 54 0 + 9 + 24 n 4 3 n 1 6 + =/
S 2 9 12 39 9= ^ + h S 2 10 9 36 10= ^ + h S 2 10 12 39 10= ^ + h j 4 j 1 6 =/
j 5 9 j 4 10 -=/
j 3 j 3 0 5 =/
n 1 n 0 100 -=/
S 2 9 12 36 9= ^ + h k 3 2 k 2 7 -=/
3k 4 0 7 +/
3k 2 n 2 7 -=/
7 + 14 + 21 + 28 + ... + 70 1000 + 999 + 998 + 997 + 996 + 995+ ... + 1 -30 - 20 - 10 - 0 + 10 + 20 + 30 + 40 a a a a b b b b c c c c d d d dA tall fence has the shape of a trapezium and has planks arranged as shown. The difference between the lengths of adjacent planks is a constant and so the lengths of the planks form an arithmetic sequence. The shortest plank is 180 cm in length and the longest string is 250 cm. The sum of the lengths of the planks is 774 m
Sequences and Series
Answers
20. c T3=30 b b c c S25=900 S11=253 S25=575 S10=-405 425 S25 =-600 S13 =-19. a a 21. b d c S100=10 100 S16=768 S201=19 899 S33=1914 a 22. b c S20=2000 a 23. T2=-8 24. 25. T5=4 d=-24 a=100 b c a a=1735,d=-6,Tn=-6n+1750 n=291 S291=254 334 26. b e c f g a Tn=3n 45 19 170 is not a multiple of 3 Yes. d = 3 n = 14 n=35 b 28. 30. 27. a T6 = $48 750 S5 = $232 500 chairs 864 S27= 29. S40 = $27 500 000 31. a b 7 0 7 1 7 2 7 3 7 4 7 5 105 + + + + + = ^ h ^ h ^ h ^ h ^ h ^ h 5 14 23 32 41 50 59 68 292 - - - - =-a a a a a a c e c b d b b b b b d fThe number of terms in the sequence is 6 The number of terms in the sequence is 7
T1=3 T2=6 T3=1 T4=-7 T1=9 T2=6 T3=2 T4=-3 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. T5=140 n=21 n=15 T5=-55 7, 14, 21, 28, 35 -2, -5, -8, -11, -14 13, 19, 25, 31, 37 -19, -38, -57, -76, -95 1. 2. 3. Tn=2n Tn=3n+1 T3=13 T4=18 T5=23 T2=13 T3=9 T4=5 T2=12 T7=-21 T3=-5 T4=-9 T5=-13 T2=1 T3=-9 T4=-4 T5=1 a a a a c b b d b b d = 4 Tn=12n-3 5 Tn=- n+26 T20=-74 a = 21 d =-5 d=4 d=-11 (i) 6 (ii) T1=12 (iii) T5=28 (iv) d = 4 (i) 5 (ii) T1=1 (iii) T5=25 (iv) d = 6 (i) 11 (ii) T1=0 (iii) T5=12 (iv) d = 3 (i) 7 (ii) T1=-10 (iii) T5=2 (iv) d = 3 (i) 6 (ii) T1=-5 (iii) T5=43 (iv) d = 12 (i) 6 (ii) T1=4 2. (iii) T5=10 6. (iv) d = 1.6 16. a = 3 33. a S7=287 b S7=-155 664 32. a 6n b n 1 6 =
/
7n 5 n 1 5 -=/
Basics
More Questions
Knowing More
Using Our Knowledge
Sequences and Series
Answers
3. 4. 5. a a c a a a a a a a a a a a b b d b b b b b b b b c c c b b 6. 4, 8, 12, 16, 20 8, 7, 6, 5, 4 8, 13, 18, 23, 28 -21, -28, -35, -42, -49 6 terms. T1 = 3, T3 = 1, T6 = 45 7 terms. T1 = 9, T3 = 2, T6 = 20 7. c 18 a -8 b 8. T5=-24 T5=63 9. n=34 n=21 10. Tn=20n-18 T n 7 n= c c Tn=5n-11 11. , , 26 18 10 13 T2= T3=0 , , 14 21 28 -12,-5 2, , , 27 35 43 - - -12. 13. 14. d=14 15. T7=-35 Tn=21-8n 16. 17. Tn=13n-1 T20=259 18. 19. a = 12, d = 13 a = 12, d = 13, Tn=4n+9 a = 4, d = 2 d = 5, x = 25 d = 10, x = 7 53 -d = 3, x = 5 d = -1, x = 2 d = -12 d = 9 a = 24 20. 21. Tn=23-5n22. No, 35 is not a term in the sequence. 23. n=43 24. 18, 25, 32, 39, 46 25. 27. 28. 29. 30. 15, 22 c S 2 47 4 42 47= ^- + h a 31. 32. 33. 34. T3=77 T2=-12 , T2=1 T3=-4 35. S25=950 S25=-325 S25=450 -84 a -27 a 36. S1000=500 500 c S16=-1720 b S16=-664 Sn=201 483 Sn=-33 540 a b 37. S41=820 c Sn=870 d Sn=-23 862 38. 39. T8=-934 a a a b b b c c 40. T4=-14 d=13 5 a=- 3 41. n=1651 S1651=5 459 763 S9=270 42. Yes, d = 6 n = 29 n = 51 e f c g Tn=6n 695 is not a multiple of 6 a = 6, d = -4 b 43. S25=2925 44. S100=10 000 45. S14=2793 46. 47. S nx n 2 1 n= ^ + h S22=528 2 48. 49. 50. 147 d 64, 128, 192 d 51. 52. 53. $38 570 a -3, 27 b 4 + 6 + ... + 44 b a b 54. $35 750 S7=$250 250 chairs S24=1392 55. 56. $960 000 $ S26= 263 250 000 a b 57. T9=$7 200 000
