Section 6-3 Arithmetic and Geometric Sequences

Section 6-3 Arithmetic and Geometric Sequences

Arithmetic and Geometric Sequences

nth-Term Formulas

Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite Geometric Series Sum Formula for Infinite Geometric Series

For most sequences it is difficult to sum an arbitrary number of terms of the sequence without adding term by term. But particular types of sequences, arith-metic sequences and geometric sequences, have certain properties that lead to con-venient and useful formulas for the sums of the corresponding arithmetic series and geometric series.

Arithmetic and Geometric Sequences

The sequence 5, 7, 9, 11, 13, . . . , 5 ⫹ 2(n ⫺ 1), . . . , where each term after the first is obtained by adding 2 to the preceding term, is an example of an arithmetic sequence. The sequence 5, 10, 20, 40, 80, . . . , 5 (2)n⫺1, . . . , where each term after the first is obtained by multiplying the preceding term by 2, is an example of a geometric sequence.

ARITHMETIC SEQUENCE

A sequence

a1, a2, a3, . . . , an, . . .

is called an arithmetic sequence, or arithmetic progression, if there exists a constant d, called the common difference, such that

an ⫺ an⫺1 ⫽ d That is, an ⫽ an⫺1 ⫹ d for every n⬎ 1 GEOMETRIC SEQUENCE A sequence a1, a2, a3, . . . , an, . . .

is called a geometric sequence, or geometric progression, if there exists a nonzero constant r, called the common ratio, such that

D E F I N I T I O N

1

D E F I N I T I O N

E x p l o r e / D i s c u s s

1

That is,

an⫽ ran⫺1 for every n ⬎ 1

(A) Graph the arithmetic sequence 5, 7, 9, . . . .

Describe the graphs of all arithmetic sequences with common difference 2.

(B) Graph the geometric sequence 5, 10, 20, . . . .

Describe the graphs of all geometric sequences with common ratio 2.

Recognizing Arithmetic and Geometric Sequences

Which of the following can be the first four terms of an arithmetic sequence? Of a geometric sequence?

(A) 1, 2, 3, 5, . . . (B) ⫺1, 3, ⫺9, 27, . . . (C) 3, 3, 3, 3, . . . (D) 10, 8.5, 7, 5.5, . . .

S o l u t i o n s (A) Since 2 ⫺ 1 ⫽ 5 ⫺ 3, there is no common difference, so the sequence is

not an arithmetic sequence. Since , there is no common ratio, so the sequence is not geometric either.

(B) The sequence is geometric with common ratio ⫺3, but it is not arithmetic. (C) The sequence is arithmetic with common difference 0 and it is also geometric

with common ratio 1.

(D) The sequence is arithmetic with common difference ⫺1.5, but it is not geometric.

Which of the following can be the first four terms of an arithmetic sequence? Of a geometric sequence?

(A) 8, 2, 0.5, 0.125, . . . (B) ⫺7, ⫺2, 3, 8, . . . (C) 1, 5, 25, 100, . . .

nth-Term Formulas

If {an} is an arithmetic sequence with common difference d, then a2 ⫽ a1 ⫹ d a3 ⫽ a2 ⫹ d ⫽ a1 ⫹ 2d a4 ⫽ a3 ⫹ d ⫽ a1 ⫹ 3d M A T C H E D P R O B L E M

1

2 1⫽ 3 2 E X A M P L E

1

an an⫺1 ⫽ r

D E F I N I T I O N

2

continued

T H E O R E M

2

T H E O R E M

1

This suggests Theorem 1, which can be proved by mathematical induction (see Problem 63 in Exercise 6-3).

THE nTH TERM OF AN ARITHMETIC SEQUENCE

an ⫽ a1 ⫹ (n ⫺ 1)d for every n ⬎ 1

Similarly, if {an} is a geometric sequence with common ratio r, then a2⫽ a1r

a3⫽ a2r ⫽ a1r 2

a4⫽ a3r ⫽ a1r 3

This suggests Theorem 2, which can also be proved by mathematical induction (see Problem 69 in Exercise 6-3).

THE nTH TERM OF A GEOMETRIC SEQUENCE

an ⫽ a1r

n⫺1 _{for every n ⬎ 1}

Finding Terms in Arithmetic and Geometric Sequences

(A) If the first and tenth terms of an arithmetic sequence are 3 and 30, respec-tively, find the fiftieth term of the sequence.

(B) If the first and tenth terms of a geometric sequence are 1 and 4, find the seventeenth term to three decimal places.

S o l u t i o n s (A) First use Theorem 1 with a1 ⫽ 3 and Now find a50:

a10⫽ 30 to find d:

a50 ⫽ a1 ⫹ (50 ⫺ 1)3

⫽ 3 ⫹ 49 ⴢ 3 ⫽ 150

(B) First let n ⫽ 10, a1 ⫽ 1, a10 ⫽ 4 and use Theorem 2 to find r. an ⫽ a1r n⫺1 4 ⫽ 1r10⫺1 r ⫽ 41/9 d⫽ 3 30⫽ 3 ⫹ 9d a10⫽ a1⫹ (10 ⫺ 1)d an⫽ a1⫹ (n ⫺ 1)d E X A M P L E

2

T H E O R E M

4

T H E O R E M

3

Now use Theorem 2 again, this time with n ⫽ 17. a17⫽ a1r

17 _{⫽ 1 (4}1/9

)17 _{⫽ 4}17/9_{艐 13.716}

(A) If the first and fifteenth terms of an arithmetic sequence are ⫺5 and 23, respectively, find the seventy-third term of the sequence.

(B) Find the eighth term of the geometric sequence

Sum Formulas for Finite Arithmetic Series

If a1, a2, a3, . . . , anis a finite arithmetic sequence, then the corresponding series a1⫹ a2 ⫹ a3 ⫹ . . . ⫹ anis called an arithmetic series. We will derive two sim-ple and very useful formulas for the sum of an arithmetic series. Let d be the common difference of the arithmetic sequence a1, a2, a3, . . . , anand let Sndenote the sum of the series a1 ⫹ a2 ⫹ a3 ⫹ . . . ⫹ an.

Then

Sn ⫽ a1 ⫹ (a1⫹ d) ⫹ . . . ⫹ [a1⫹ (n ⫺ 2)d] ⫹ [a1 ⫹ (n ⫺ 1)d]

Reversing the order of the sum, we obtain

Sn ⫽ [a1⫹ (n ⫺ 1)d] ⫹ [a1 ⫹ (n ⫺ 2)d] ⫹ . . . ⫹ (a1 ⫹ d) ⫹ a1

Adding the left sides of these two equations and corresponding elements of the right sides, we see that

2Sn ⫽ [2a1 ⫹ (n ⫺ 1)d] ⫹ [2a1⫹ (n ⫺ 1)d] ⫹ . . . ⫹ [2a1⫹ (n ⫺ 1)d]

⫽ n[2a1 ⫹ (n ⫺ 1)d]

This can be restated as in Theorem 3.

SUM OF AN ARITHMETIC SERIES——FIRST FORM

Sn⫽ [2a1 ⫹ (n ⫺ 1)d]

By replacing a1 ⫹ (n ⫺ 1)d with an, we obtain a second useful formula for the sum.

SUM OF AN ARITHMETIC SERIES——SECOND FORM

Sn⫽ (a1⫹ an) n 2 n 2 1 64, ⫺ 1 32, 1 16, . . . . M A T C H E D P R O B L E M

2

The proof of the first sum formula by mathematical induction is left as an exercise (see Problem 64 in Exercise 6-3).

Finding the Sum of an Arithmetic Series

Find the sum of the first 26 terms of an arithmetic series if the first term is

⫺7 and d ⫽ 3.

S o l u t i o n Let n ⫽ 26, a1⫽ ⫺7, d ⫽ 3, and use Theorem 3.

Sn ⫽ [2a1⫹ (n ⫺ 1)d] S26⫽ [2(⫺7) ⫹ (26 ⫺ 1)3]

⫽ 793

Find the sum of the first 52 terms of an arithmetic series if the first term is 23 and d ⫽ ⫺2.

Finding the Sum of an Arithmetic Series

Find the sum of all the odd numbers between 51 and 99, inclusive.

S o l u t i o n First, use a1 ⫽ 51, an ⫽ 99, and Now use Theorem 4 to find S25:

Theorem 1 to find n:

an ⫽ a1 ⫹ (n ⫺ 1)d Sn ⫽ (a1⫹ an)

99 ⫽ 51 ⫹ (n ⫺ 1)2 S25⫽ (51 ⫹ 99)

n ⫽ 25 ⫽ 1,875

Find the sum of all the even numbers between ⫺22 and 52, inclusive.

Prize Money

A 16-team bowling league has $8,000 to be awarded as prize money. If the last-place team is awarded $275 in prize money and the award increases by the same amount for each successive finishing place, how much will the first-place team receive?

E X A M P L E

5

M A T C H E D P R O B L E M

4

25 2 n 2 E X A M P L E

4

M A T C H E D P R O B L E M

3

26 2 n 2 E X A M P L E

3

T H E O R E M

5

S o l u t i o n If a1 is the award for the first-place team, a2 is the award for the second-place

team, and so on, then the prize money awards form an arithmetic sequence with n⫽ 16, a16 ⫽ 275, and S16 ⫽ 8,000. Use Theorem 4 to find a1.

Sn⫽ (a1 ⫹ an) 8,000 ⫽ (a1⫹ 275)

a1⫽ 725

Thus, the first-place team receives $725.

Refer to Example 5. How much prize money is awarded to the second-place team?

Sum Formulas for Finite Geometric Series

If a1, a2, a3, . . . , anis a finite geometric sequence, then the corresponding series a1⫹ a2⫹ a3 ⫹ . . . ⫹ anis called a geometric series. As with arithmetic series, we can derive two simple and very useful formulas for the sum of a geometric series. Let r be the common ratio of the geometric sequence a1, a2, a3, . . . , an and let Sn denote the sum of the series a1 ⫹ a2⫹ a3⫹ . . . ⫹ an. Then

Sn ⫽ a1 ⫹ a1r⫹ a1r 2 _{⫹ a} 1r 3 _{⫹ . . . ⫹ a} 1r n⫺2_{⫹ a} 1r n⫺1

Multiply both sides of this equation by r to obtain rSn ⫽ a1r⫹ a1r 2 _{⫹ a} 1r 3 _{⫹ . . . ⫹ a} 1r n⫺1_{⫹ a} 1r n

Now subtract the left side of the second equation from the left side of the first, and the right side of the second equation from the right side of the first to obtain

Sn ⫺ rSn ⫽ a1⫺ a1r n

Sn(1 ⫺ r) ⫽ a1 ⫺ a1r n

Thus, solving for Sn, we obtain the following formula for the sum of a geomet-ric series:

SUM OF A GEOMETRIC SERIES——FIRST FORM

Sn⫽ r ⫽ 1 a1⫺ a1r n 1⫺ r M A T C H E D P R O B L E M

5

16 2 n 2

T H E O R E M

6

Since an ⫽ a1r n⫺1 , or ran⫽ a1r n

, the sum formula also can be written in the following form:

SUM OF A GEOMETRIC SERIES——SECOND FORM

Sn ⫽ r⫽ 1

The proof of the first sum formula (Theorem 5) by mathematical induction is left as an exercise (see Problem 70, Exercise 6-3).

If r⫽ 1, then Sn⫽ a1⫹ a1(1) ⫹ a1(1 2 ) ⫹ . . . ⫹ a1(1 n⫺1 ) ⫽ na1

Finding the Sum of a Geometric Series

Find the sum of the first 20 terms of a geometric series if the first term is 1 and r ⫽ 2.

S o l u t i o n Let n ⫽ 20, a1⫽ 1, r ⫽ 2, and use Theorem 5.

Calculation using a calculator

Find the sum, to two decimal places, of the first 14 terms of a geometric series if the first term is and r ⫽ ⫺2.

Sum Formula for Infinite Geometric Series

Consider a geometric series with a1⫽ 5 and r ⫽ . What happens to the sum Sn as n increases? To answer this question, we first write the sum formula in the more convenient form

(1) For a1⫽ 5 and r ⫽ , Sn⫽ 10 ⫺ 10

冢

1 2

冣

n 1 2 Sn⫽ a1⫺ a1r n 1⫺ r ⫽ a1 1⫺ r⫺ a1r n 1⫺ r 1 2 1 64 M A T C H E D P R O B L E M

6

⫽1⫺ 1 ⴢ 2 20 1⫺ 2 ⫽ 1,048,575 Sn⫽ a1⫺ a1r n 1⫺ r E X A M P L E

6

a1⫺ ran 1⫺ r

Thus,

It appears that becomes smaller and smaller as n increases and that the sum gets closer and closer to 10.

In general, it is possible to show that, if ⬍ 1, then rn

will get closer and closer to 0 as n increases. Symbolically, rn →

0 as n→⬁. Thus, the term

in equation (1) will tend to 0 as n increases, and Sn will tend to

In other words, if ⬍ 1, then Sn can be made as close to

as we wish by taking n sufficiently large. Thus, we define the sum of an infinite

geometric series by the following formula:

SUM OF AN INFINITE GEOMETRIC SERIES

S⬁⫽

If ⱖ 1, an infinite geometric series has no sum.

Expressing a Repeating Decimal as a Fraction

Represent the repeating decimal 0.454 545 . . .⫽ as the quotient of two integers. Recall that a repeating decimal names a rational number and that any rational number can be represented as the quotient of two integers.

0.45 E X A M P L E

7

ⱍ

r

ⱍ

a1 1⫺ r

ⱍ

r

ⱍ

⬍ 1 a1 1⫺ r

ⱍ

r

ⱍ

a1 1⫺ r a1r n 1⫺ r

ⱍ

r

ⱍ

(1₂)n S20⫽ 10 ⫺ 10

冢

1 1,048,576

冣

S10⫽ 10 ⫺ 10

冢

1 1,024

冣

S4⫽ 10 ⫺ 10

冢

1 16

冣

S2⫽ 10 ⫺ 10

冢

1 4

冣

D E F I N I T I O N

3

E x p l o r e / D i s c u s s

2

S o l u t i o n ⫽ 0.45 ⫹ 0.0045 ⫹ 0.000 045 ⫹ . . .

The right side of the equation is an infinite geometric series with a1 ⫽ 0.45 and r ⫽ 0.01. Thus,

S_⬁⫽

Hence, and name the same rational number. Check the result by dividing 5 by 11.

Repeat Example 7 for 0.818 181 . . .⫽ .

Economy Stimulation

A state government uses proceeds from a lottery to provide a tax rebate for property owners. Suppose an individual receives a $500 rebate and spends 80% of this, and each of the recipients of the money spent by this individual also spends 80% of what he or she receives, and this process continues with-out end. According to the multiplier doctrine in economics, the effect of the original $500 tax rebate on the economy is multiplied many times. What is the total amount spent if the process continues as indicated?

S o l u t i o n The individual receives $500 and spends 0.8(500) ⫽ $400. The recipients of this

$400 spend 0.8(400) ⫽ $320, the recipients of this $320 spend 0.8(320) ⫽ $256, and so on. Thus, the total spending generated by the $500 rebate is

400 ⫹ 320 ⫹ 256 ⫹ . . . ⫽ 400 ⫹ 0.8(400) ⫹ (0.8)2

(400) ⫹ . . .

which we recognize as an infinite geometric series with a1 ⫽ 400 and r ⫽ 0.8.

Thus, the total amount spent is

S_⬁⫽

Repeat Example 8 if the tax rebate is $1,000 and the percentage spent by all recip-ients is 90%.

(A) Find an infinite geometric series with a1 ⫽ 10 whose sum is 1,000.

(B) Find an infinite geometric series with a1 ⫽ 10 whose sum is 6.

(C) Suppose that an infinite geometric series with a1 ⫽ 10 has a sum.

Explain why that sum must be greater than 5.

M A T C H E D P R O B L E M

8

a1 1⫺ r⫽ 400 1⫺ 0.8⫽ 400 0.2 ⫽ $2,000 E X A M P L E

8

0.81 M A T C H E D P R O B L E M

7

5 11 0.45 a1 1⫺ r⫽ 0.45 1⫺ 0.01⫽ 0.45 0.99⫽ 5 11 0.45

A n s w e r s t o M a t c h e d P r o b l e m s

1. (A) The sequence is geometric with r⫽ , but not arithmetic.

(B) The sequence is arithmetic with d⫽ 5, but not geometric. (C) The sequence is neither arithmetic nor geometric.

2. (A) 139 (B) ⫺2 3. ⫺1,456 4. 570 5. $695 6. ⫺85.33 7. ₁₁9 8. $9,000

1 4

E X E R C I S E 6 - 3

A

In Problems 1 and 2, determine whether the following can be the first three terms of an arithmetic or geometric sequence, and, if so, find the common difference or common ratio and the next two terms of the sequence.

1. (A) ⫺11, ⫺16, ⫺21, . . . (B) 2, ⫺4, 8, . . .

(C) 1, 4, 9, . . . (D) , . . .

2. (A) 5, 20, 100, . . . (B) ⫺5, ⫺5, ⫺5, . . . (C) 7, 6.5, 6, . . . (D) 512, 256, 128, . . .

Let a1, a2, a3, . . . , an, . . . be an arithmetic sequence. In

Problems 3–10, find the indicated qualities.

3. a1⫽ ⫺5, d ⫽ 4; a2⫽ ?, a3⫽ ?, a4⫽ ? 4. a1⫽ ⫺18, d ⫽ 3; a2⫽ ?, a3⫽ ?, a4⫽ ? 5. a1⫽ ⫺3, d ⫽ 5; a15⫽ ?, S11⫽ ? 6. a1⫽ 3, d ⫽ 4; a22⫽ ?, S21⫽ ? 7. a1⫽ 1, a2⫽ 5; S21⫽ ? 8. a1⫽ 5, a2⫽ 11; S11⫽ ? 9. a1⫽ 7, a2⫽ 5; a15⫽ ? 10. a1⫽ ⫺3, d ⫽ ⫺4; a10⫽ ?

Let a1, a2, a3, . . . , an, . . . be a geometric sequence. In

Problems 11–16, find each of the indicated quantities.

11. a1⫽ ⫺6, r ⫽ ⫺ ; a2⫽ ?, a3⫽ ?, a4⫽ ? 12. a1⫽ 12, r ⫽ ; a2⫽ ?, a3⫽ ?, a4⫽ ? 13. a1⫽ 81, r ⫽ ; a10⫽ ? 14. a1⫽ 64, r ⫽ ; a13⫽ ? 15. a1⫽ 3, a7⫽ 2,187, r ⫽ 3; S7⫽ ? 16. a1⫽ 1, a7⫽ 729, r ⫽ ⫺3; S7⫽ ?

B

Let a1, a2, a3, . . . , an, . . . be an arithmetic sequence. In

Problems 17–24, find the indicated quantities.

1 2 1 3 2 3 1 2 1 2, 1 6, 1 18 17. a1⫽ 3, a20⫽ 117; d ⫽ ?, a101⫽ ? 18. a1⫽ 7, a8⫽ 28; d ⫽ ?, a25⫽ ? 19. a1⫽ ⫺12, a40⫽ 22; S40⫽ ? 20. a1⫽ 24, a24⫽ ⫺28; S24⫽ ? 21. a1⫽ , a2⫽ ; a11⫽ ?, S11⫽ ? 22. a1⫽ , a2⫽ ; a19⫽ ?, S19⫽ ? 23. a3⫽ 13, a10⫽ 55; a1⫽ ? 24. a9⫽ ⫺12, a13⫽ 3; a1⫽ ?

Let a1, a2, a3, . . . , an, . . . be a geometric sequence. Find each

of the indicated quantities in Problems 25–30.

25. a1⫽ 100, a6⫽ 1; r ⫽ ? 26. a1⫽ 10, a10⫽ 30; r ⫽ ? 27. a1⫽ 5, r ⫽ ⫺2; S10⫽ ? 28. a1⫽ 3, r ⫽ 2; S10⫽ ? 29. a1⫽ 9, a4⫽ ; a2⫽ ?, a3⫽ ? 30. a1⫽ 12, a4⫽ ⫺ ; a2⫽ ?, a3⫽ ? 31. S51⫽ ⫽ ? 32. S40⫽ ⫽ ? 33. S7⫽ ⫽ ? 34. S7⫽ ⫽ ? 35. Find g(1) ⫹ g(2) ⫹ g(3) ⫹ . . . ⫹ g(51) if g(t) ⫽ 5 ⫺ t. 36. Find f (1) ⫹ f(2) ⫹ f(3) ⫹ . . . ⫹ f(20) if f(x) ⫽ 2x ⫺ 5. 37. Find g(1) ⫹ g(2) ⫹ . . . ⫹ g(10) if g(x) ⫽ . 38. Find f (1) ⫹ f(2) ⫹ . . . ⫹ f(10) if f(x) ⫽ 2x .

39. Find the sum of all the even integers between 21 and 135. 40. Find the sum of all the odd integers between 100 and 500. 41. Show that the sum of the first n odd natural numbers is n2

, using approximate formulas from this section.

42. Show that the sum of the first n even natural numbers is

n⫹ n2

, using appropriate formulas from this section.

43. Find a positive number x so that ⫺2 ⫹ x ⫺ 6 is a

three-term geometric series.

44. Find a positive number x so that 6 ⫹ x ⫹ 8 is a three-term

geometric series.

45. For a given sequence in which a1⫽ ⫺3 and an⫽ an⫺1⫹ 3,

n⬎ 1, find anin terms of n. (1₂)x

兺

7 k⫽1 3k

兺

7 k⫽1 (⫺3)k⫺1

兺

40 k⫽1 (2k⫺ 3)

兺

51 k⫽1 (3k⫹ 3) 4 9 8 3 1 4 1 6 1 2 1 3

46. For the sequence in Problem 45, find Sn⫽ akin terms

of n.

In Problems 47–50, find the least positive integer n such that an⬍ bnby graphing the sequences {an} and {bn} with a

graphing utility. Check your answer by using a graphing utility to display both sequences in table form.

47. an⫽ 5 ⫹ 8n, bn⫽ 1.1 n 48. an⫽ 96 ⫹ 47n, bn⫽ 8(1.5) n 49. an⫽ 1,000 (0.99) n , bn⫽ 2n ⫹ 1 50. an⫽ 500 ⫺ n, bn⫽ 1.05 n

In Problems 51–56, find the sum of each infinite geometric series that has a sum.

51. 3 ⫹ 1 ⫹ ⫹ . . . 52. 16 ⫹ 4 ⫹ 1 ⫹ . . . 53. 2 ⫹ 4 ⫹ 8 ⫹ . . . 54. 4 ⫹ 6 ⫹ 9 ⫹ . . . 55. 2 ⫺ ⫹ ⫺ . . . 56. 21 ⫺ 3 ⫹ ⫺ . . .

In Problems 57–62, represent each repeating decimal as the quotient of two integers.

57. ⫽ 0.7777 . . . 58. ⫽ 0.5555 . . . 59. ⫽ 0.545 454 . . . 60. ⫽ 0.272 727 . . . 61. ⫽ 3.216 216 216 . . .

62. ⫽ 5.636 363 . . .

C

63. Prove, using mathemtical induction, that if {an} is an

arithmetic sequence, then

an⫽ a1⫹ (n ⫺ 1)d for every n⬎ 1

64. Prove, using mathematical induction, that if {an} is an

arithmetic sequence, then

Sn⫽ [2a1⫹ (n ⫺ 1)d]

65. If in a given sequence, a1⫽ ⫺2 and an⫽ ⫺3an⫺1, n⬎ 1,

find anin terms of n.

66. For the sequence in Problem 65, find Sn⫽ akin terms

of n.

67. Show that (x2_{⫹ xy ⫹ y}2

), (z2_{⫹ xz ⫹ x}2

), and ( y2_{⫹ yz ⫹}

z2

) are consecutive terms of an arithmetic progression if x,

y, and z form an arithmetic progression. (From U.S.S.R.

Mathematical Olympiads, 1955–1956, Grade 9.)

68. Take 121 terms of each arithmetic progression 2, 7,

12, . . . and 2, 5, 8, . . . . How many numbers will there be in common? (From U.S.S.R. Mathematical Olympiads, 1955–1956, Grade 9.)

兺

n k⫽1 n 2 5.63 3.216 0.27 0.54 0.5 0.7 3 7 1 8 1 2 1 3

兺

n k⫽1

69. Prove, using mathematical induction, that if {an} is a

geo-metric sequence, then

an⫽ a1r

n⫺1 _{n僆 N}

70. Prove, using mathematical induction, that if {an} is a

geo-metric sequence, then

Sn⫽ n僆 N, r ⫽ 1

71. Given the system of equations

ax⫹ by ⫽ c dx⫹ ey ⫽ f

where a, b, c, d, e, and f is any arithmetic progression with a nonzero constant difference, show that the system has a unique solution.

72. The sum of the first and fourth terms of an arithmetic

se-quence is 2, and the sum of their squares is 20. Find the sum of the first eight terms of the sequence.

A P P L I C AT I O N S

73. Business. In investigating different job opportunities, you

find that firm A will start you at $25,000 per year and guarantee you a raise of $1,200 each year while firm B will start you at $28,000 per year but will guarantee you a raise of only $800 each year. Over a period of 15 years, how much would you receive from each firm?

74. Business. In Problem 73, what would be your annual

salary at each firm for the tenth year?

75. Economics. The government, through a subsidy program,

distributes $1,000,000. If we assume that each individual or agency spends 0.8 of what is received, and 0.8 of this is spent, and so on, how much total increase in spending re-sults from this government action?

76. Economics. Due to reduced taxes, an individual has an

extra $600 in spendable income. If we assume that the in-dividual spends 70% of this on consumer goods, that the producers of these goods in turn spend 70% of what they receive on consumer goods, and that this process contin-ues indefinitely, what is the total amount spent on con-sumer goods?

★77. Business. If $P is invested at r% compounded annually,

the amount A present after n years forms a geometric pro-gression with a common ratio 1 ⫹ r. Write a formula for the amount present after n years. How long will it take a sum of money P to double if invested at 6% interest com-pounded annually?

★78. Population Growth. If a population of A0people grows at

the constant rate of r% per year, the population after t years forms a geometric progression with a common ratio 1 ⫹ r. Write a formula for the total population after t years. If the world’s population is increasing at the rate of 2% per year, how long will it take to double?

a1⫺ a1r n

79. Finance. Eleven years ago an investment earned $7,000

for the year. Last year the investment earned $14,000. If the earnings from the investment have increased the same amount each year, what is the yearly increase and how much income has accrued from the investment over the past 11 years?

80. Air Temperature. As dry air moves upward, it expands.

In so doing, it cools at the rate of about 5°F for each 1,000-foot rise. This is known as the adiabatic process. (A) Temperatures at altitudes that are multiples of 1,000

feet form what kind of a sequence?

(B) If the ground temperature is 80°F, write a formula for the temperature Tnin terms of n, if n is in thousands

of feet.

81. Engineering. A rotating flywheel coming to rest rotates

300 revolutions the first minute (see the figure). If in each subsequent minute it rotates two-thirds as many times as in the preceding minute, how many revolutions will the wheel make before coming to rest?

82. Physics. The first swing of a bob on a pendulum is 10

inches. If on each subsequent swing it travels 0.9 as far as on the preceding swing, how far will the bob travel before coming to rest?

83. Food Chain. A plant is eaten by an insect, an insect by a

trout, a trout by a salmon, a salmon by a bear, and the bear is eaten by you. If only 20% of the energy is transformed from one stage to the next, how many calories must be supplied by plant food to provide you with 2,000 calories from the bear meat?

★84. Genealogy. If there are 30 years in a generation, how

many direct ancestors did each of us have 600 years ago? By direct ancestors we mean parents, grandparents, great-grandparents, and so on.

★85. Physics. An object falling from rest in a vacuum near the

surface of the Earth falls 16 feet during the first second, 48 feet during the second second, 80 feet during the third sec-ond, and so on.

(A) How far will the object fall during the eleventh second?

(B) How far will the object fall in 11 seconds? (C) How far will the object fall in t seconds?

★86. Physics. In Problem 85, how far will the object fall

during:

(A) The twentieth second? (B) The t th second?

★87. Bacteria Growth. A single cholera bacterium divides

every hour to produce two complete cholera bacteria. If we start with a colony of A0bacteria, how many bacteria

will we have in t hours, assuming adequate food supply?

★88. Cell Division. One leukemic cell injected into a healthy

mouse will divide into two cells in about day. At the end of the day these two cells will divide again, with the dou-bling process continuing each day until there are 1 bil-lion cells, at which time the mouse dies. On which day after the experiment is started does this happen?

★★89. Astronomy. Ever since the time of the Greek astronomer

Hipparchus, second century B.C., the brightness of stars

has been measured in terms of magnitude. The brightest stars, excluding the sun, are classed as magnitude 1, and the dimmest visible to the eye are classed as magnitude 6. In 1856, the English astronomer N. R. Pogson showed that first-magnitude stars are 100 times brighter than sixth-magnitude stars. If the ratio of brightness between consec-utive magnitudes is constant, find this ratio. [Hint: If bnis

the brightness of an nth-magnitude star, find r for the geo-metric progression b1, b2, b3, . . . , given b1⫽ 100b6.] ★90. Music. The notes on a piano, as measured in cycles per

second, form a geometric progression.

(A) If A is 400 cycles per second and A⬘, 12 notes higher, is 800 cycles per second, find the constant ratio r.

1 2 1 2 1 2

(B) Find the cycles per second for C, three notes higher than A.

91. Puzzle. If you place 1¢ on the first square of a chessboard,

2¢ on the second square, 4¢ on the third, and so on, con-tinuing to double the amount until all 64 squares are cov-ered, how much money will be on the sixty-fourth square? How much money will there be on the whole board?

★92. Puzzle. If a sheet of very thin paper 0.001 inch thick is

torn in half, and each half is again torn in half, and this process is repeated for a total of 32 times, how high will the stack of paper be if the pieces are placed one on top of the other? Give the answer to the nearest mile.

★93. Atmospheric Pressure. If atmospheric pressure decreases

roughly by a factor of 10 for each 10-mile increase in

alti-tude up to 60 miles, and if the pressure is 15 pounds per square inch at sea level, what will the pressure be 40 miles up?

94. Zeno’s Paradox. Visualize a hypothetical 440-yard oval

racetrack that has tapes stretched across the track at the halfway point and at each point that marks the halfway point of each remaining distance thereafter. A runner run-ning around the track has to break the first tape before the second, the second before the third, and so on. From this point of view it appears that he will never finish the race. This famous paradox is attributed to the Greek philoso-pher Zeno (495–435 B.C.). If we assume the runner runs at 440 yards per minute, the times between tape breakings form an infinite geometric progression. What is the sum of this progression?

95. Geometry. If the midpoints of the sides of an equilateral

triangle are joined by straight lines, the new figure will be an equilateral triangle with a perimeter equal to half the original. If we start with an equilateral triangle with perimeter 1 and form a sequence of “nested” equilateral triangles proceeding as described, what will be the total perimeter of all the triangles that can be formed in this way?

96. Photography. The shutter speeds and f-stops on a camera

are given as follows: Shutter speeds:

f-stops: 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22 These are very close to being geometric progressions. Es-timate their common ratios.

★★97. Geometry. We know that the sum of the interior angles of

a triangle is 180°. Show that the sums of the interior an-gles of polygons with 3, 4, 5, 6, . . . sides form an arith-metic sequence. Find the sum of the interior angles for a 21-sided polygon. 1, 12, 1 4, 1 8, 1 15, 1 30, 1 60, 1 125, 1 250, 1 500

Section 6-4 Multiplication Principle, Permutations,

and Combinations

Multiplication Principle Factorial

Permutations Combinations

This section introduces some new mathematical tools that are usually referred to as counting techniques. In general, a counting technique is a mathematical method of determining the number of objects in a set without actually enumerat-ing the objects in the set as 1, 2, 3, . . . . For example, we can count the number