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11.3 Geometric Sequences and Series

Identify the common ratio of a geometric sequence, and find a given term and the sum of the first n terms.
Find the sum of an infinite geometric series, if it exists.

A sequence in which each term after the first is found by multiplying the preceding term by the same number is a geometric sequence.

Geometric Sequences

Consider the sequence:

2, 6, 18, 54, 162, \dots .

$2, 6, 18, 54, 162, \dots .$

Note that multiplying each term by 3 produces the next term. We call the number 3 the common ratio because it can be found by dividing any term by the preceding term. A geometric sequence is also called a geometric progression.

Example 1

For each of the following geometric sequences, identify the common ratio.

3, 6, 12, 24, 48, …
1, −12, 14, −18,… $1, - \frac{1}{2}, \frac{1}{4}, - \frac{1}{8}, \dots$
$5200, $3900, $2925, $2193.75,…
$1000, $1060, $1123.60,…

Solution

SEQUENCE	COMMON RATIO
a) 3, 6, 12, 24, 48, …	$2 (6 3 = 2, 12 6 = 2, and so on)$ $2 (\frac{6}{3} = 2, \frac{12}{6} = 2, and so on)$
b) 1, −12, 14, −18,… $1, - \frac{1}{2}, \frac{1}{4}, - \frac{1}{8}, \dots$	$- 1 2 (- 1 2 1 = - 1 2, 1 4 - 1 2 = - 1 2, and so on)$ $- \frac{1}{2} (\frac{- \frac{1}{2}}{1} = - \frac{1}{2}, \frac{\frac{1}{4}}{- \frac{1}{2}} = - \frac{1}{2}, and so on)$
c) $5200, $3900, $2925, $2193.75, …	$0.75 ($3900 $5200 = 0.75, $2925 $3900 = 0.75, and so on)$ $0.75 (\frac{$3900}{$5200} = 0.75, \frac{$2925}{$3900} = 0.75, and so on)$
d) $1000, $1060, $1123.60, …	$1.06 ($1060 $1000 = 1.06, $1123.60 $1060 = 1.06, and so on)$ $1.06 (\frac{$1060}{$1000} = 1.06, \frac{$1123.60}{$1060} = 1.06, and so on)$

Now Try Exercise 1.

We now find a formula for the general, or nth, term of a geometric sequence. Let a1 $a_{1}$ be the first term and r the common ratio. The first few terms are as follows:

Generalizing, we obtain the following.

Example 2

Find the 7th term of the geometric sequence 4, 20, 100, ….

Solution

We first note that

a 1 = 4 and n = 7.

$a_{1} = 4 and n = 7.$

To find the common ratio, we can divide any term (other than the first) by the preceding term. Since the second term is 20 and the first is 4, we get

r = 20 4, or 5.

$r = \frac{20}{4}, or 5.$

Then using the formula an=a1rn−1 $a_{n} = a_{1} r^{n - 1}$ , we have

a 7 = 4 \cdot 5 7 - 1 = 4 \cdot 56 = 4 \cdot 15,625 = 62,500.

$a_{7} = 4 \cdot 5^{7 - 1} = 4 \cdot 5^{6} = 4 \cdot 15,625 = 62,500.$

Thus the 7th term is 62,500.

Now Try Exercise 11.

Example 3

Find the 10th term of the geometric sequence 64,−32, 16 $64, - 32, 16$ ,−8,… $- 8, \dots$ .

Solution

We first note that

a 1 = 64, n = 10, and r = - 32 64, or - 1 2 .

$a_{1} = 64, n = 10, and r = \frac{- 32}{64}, or - \frac{1}{2} .$

Then using the formula an=a1rn−1 $a_{n} = a_{1} r^{n - 1}$ , we have

a 10 = 64 \cdot (- 1 2) 10 - 1 = 64 \cdot (- 1 2) 9 = 26 \cdot (- 1 2 9) = - 1 2 3 = - 1 8 .

$a_{10} = 64 \cdot {(- \frac{1}{2})}^{10 - 1} = 64 \cdot {(- \frac{1}{2})}^{9} = 2^{6} \cdot (- \frac{1}{2^{9}}) = - \frac{1}{2^{3}} = - \frac{1}{8} .$

Thus the 10th term is −18 $- \frac{1}{8}$ .

Now Try Exercise 15.

Sum of the First `n` Terms of a Geometric Sequence

Next, we develop a formula for the sum Sn $S_{n}$ of the first n terms of a geometric sequence:

a 1, a 1 r, a 1 r 2, a 1 r 3, \dots, a 1 r n - 1, \dots .

$a_{1}, a_{1} r, a_{1} r^{2}, a_{1} r^{3}, \dots, a_{1} r^{n - 1}, \dots .$

The associated geometric series is given by

S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + \dots + a 1 r n - 1 .

$S_{n} = a_{1} + a_{1} r + a_{1} r^{2} + a_{1} r^{3} + \dots + a_{1} r^{n - 1} .$ (1)

We want to find a formula for this sum. If we multiply by r on both sides of equation (1), we have

r S n = a 1 r + a 1 r 2 + a 1 r 3 + a 1 r 4 + \dots + a 1 r n .

$r S_{n} = a_{1} r + a_{1} r^{2} + a_{1} r^{3} + a_{1} r^{4} + \dots + a_{1} r^{n} .$ (2)

Subtracting equation (2) from equation (1), we see that the differences of the terms shown in red are 0, leaving

S n - r S n = a 1 - a 1 r n,

$S_{n} - r S_{n} = a_{1} - a_{1} r^{n},$

S n (1 - r) = a 1 (1 - r n) . F a c t o r i n g

$\begin{array}{l} S_{n} (1 - r) = a_{1} (1 - r^{n}) . & Factoring \end{array}$

Dividing by 1−r $1 - r$ on both sides gives us the following formula.

Example 4

Find the sum of the first 7 terms of the geometric sequence 3, 15, 75, 375,….

Solution

We first note that

a 1 = 3, n = 7, and r = 15 3, or 5.

$a_{1} = 3, n = 7, and r = \frac{15}{3}, or 5.$

Then using the formula

S n = a 1 ( 1 - r n ) 1 - r,

$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r},$

we have

S 7 = = = 3 ( 1 - 5 7 ) 1 - 5 3 ( 1 - 78 , 125 ) - 4 58, 593.

$\begin{array}{rcl} S_{7} & = & \frac{3 (1 - 5^{7})}{1 - 5} \\ = & \frac{3 (1 - 78, 125)}{- 4} \\ = & 58, 593. \end{array}$

Thus the sum of the first 7 terms is 58,593.

Now Try Exercise 23.

Example 5

Find the sum: ∑k=111(0.3)k $\sum_{k = 1}^{11} {(0.3)}^{k}$ .

Solution

This is a geometric series with a1=0.3, r=0.3 $a_{1} = 0.3, r = 0.3$ , and n=11 $n = 11$ . Thus,

S 11 = \approx 0.3 ( 1 - 0.3 11 ) 1 - 0.3 0.42857.

$\begin{array}{rcl} S_{11} & = & \frac{0.3 (1 - {0.3}^{11})}{1 - 0.3} \\ \approx & 0.42857. \end{array}$

Now Try Exercise 41.

Infinite Geometric Series

The sum of the terms of an infinite geometric sequence is an infinite geometric series. For some geometric sequences, Sn $S_{n}$ gets close to a specific number as n gets large. For example, consider the infinite series

1 2 + 1 4 + 1 8 + 1 16 + \dots + 1 2 n + \dots .

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots + \frac{1}{2^{n}} + \dots .$

We can visualize Sn $S_{n}$ by considering the area of a square. For S1 $S_{1}$ , we shade half the square. For S2 $S_{2}$ , we shade half the square plus half the remaining half, or 14 $\frac{1}{4}$ . For S3 $S_{3}$ , we shade the parts shaded in S2 $S_{2}$ plus half the remaining part. We see that the values of Sn $S_{n}$ will get close to 1 (shading the complete square).

We examine some partial sums. Note that each of the partial sums in Table 1 is less than 1, but Sn $S_{n}$ gets very close to 1 as n gets large. We say that 1 is the limit of Sn $S_{n}$ and also that 1 is the sum of the infinite geometric sequence. The sum of an infinite geometric sequence is denoted S∞ $S_{\infty}$ . In this case, S∞=1 $S_{\infty} = 1$ .

Table 1

`n`	Sn $S_{n}$
1	0.5
5	0.96875
10	0.9990234375
20	0.9999990463
30	0.9999999991

Some infinite sequences do not have sums. Consider the infinite geometric series

2 + 4 + 8 + 16 + \dots + 2 n + \dots .

$2 + 4 + 8 + 16 + \dots + 2^{n} + \dots^{} .$

We again examine some partial sums. Note in Table 2 that as n gets large, Sn $S_{n}$ gets large without bound. This sequence does not have a sum.

Table 2

`n`	Sn $S_{n}$
1	2
5	62
10	2,046
20	2,097,150
30	2,147,483,646

It can be shown (but we will not do so here) that the sum of an infinite geometric series exists if and only if |r|<1 $| r | < 1$ (that is, the absolute value of the common ratio is less than 1).

To find a formula for the sum of an infinite geometric series, we first consider the sum of the first n terms:

S n = a 1 ( 1 - r n ) 1 - r = a 1 - a 1 r n 1 - r . U s i n g t h e d i s t r i b u t i v e l a w

$\begin{array}{rcl} S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} = \frac{a_{1} - a_{1} r^{n}}{1 - r} . & Using the distributive law \end{array}$

For |r|<1 $| r | < 1$ , values of rn $r^{n}$ get close to 0 as n gets large. As rn $r^{n}$ gets close to 0, so does a1rn $a_{1} r^{n}$ . Thus, Sn $S_{n}$ gets close to a1/(1−r) $a_{1} / (1 - r)$ .

Example 6

Determine whether each of the following infinite geometric series has a limit. If a limit exists, find it.

1+3+9+27+⋯ $1 + 3 + 9 + 27 + \dots$
−2+1−12+14−18+⋯ $- 2 + 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dots$

Solution

Here r=3 $r = 3$ , so |r|=|3|=3 $| r | = | 3 | = 3$ . Since |r|>1 $| r | > 1$ , the series does not have a limit.
Here r=−12 $r = - \frac{1}{2}$ , so ∣∣r∣∣=∣∣−12∣∣=12 $| r | = | - \frac{1}{2} | = \frac{1}{2}$ . Since |r|<1 $| r | < 1$ , the series does have a limit. We find the limit:

$S \infty = a 1 1 - r = - 2 1 - ( - 1 2 ) = - 2 3 2 = - 4 3 .$ $S_{\infty} = \frac{a_{1}}{1 - r} = \frac{- 2}{1 - (- \frac{1}{2})} = \frac{- 2}{\frac{3}{2}} = - \frac{4}{3} .$

Now Try Exercises 33 and 37.

Example 7

Find fraction notation for 0.78787878 …, or 0.78¯¯¯¯ $0. \bar{78}$ .

Solution

We can express this as

0.78 + 0.0078 + 0.000078 + \dots .

$0.78 + 0.0078 + 0.000078 + \dots .$

Then we see that this is an infinite geometric series, where a1=0.78 $a_{1} = 0.78$ and r=0.01 $r = 0.01$ . Since |r|<1 $| r | < 1$ , this series has a limit:

S \infty = a 1 1 - r = 0.78 1 - 0.01 = 0.78 0.99 = 78 99, or 26 33 .

$S_{\infty} = \frac{a_{1}}{1 - r} = \frac{0.78}{1 - 0.01} = \frac{0.78}{0.99} = \frac{78}{99}, or \frac{26}{33} .$

Thus fraction notation for 0.78787878…is 2633 $\frac{26}{33}$ . You can check this on your calculator.

Now Try Exercise 51.

Applications

The translation of some applications and problem-solving situations may involve geometric sequences or series. Examples 9 and 10, in particular, show applications in business and economics.

Example 8

A Daily Doubling Salary. Suppose someone offered you a job for the month of September (30 days) under the following conditions. You will be paid $0.01 for the first day, $0.02 for the second, $0.04 for the third, and so on, doubling your previous day’s salary each day. How much would you earn altogether for the month? (Would you take the job? Make a conjecture before reading further.)

Solution

You earn $0.01 the first day, $0.01(2) the second day, $0.01(2)(2) the third day, and so on. The amount earned is the geometric series

$0.01 + $0.01 (2) + $0.01 (22) + $0.01 (23) + \dots + $0.01 (229),

$$0.01 + $0.01 (2) + $0.01 (2^{2}) + $0.01 (2^{3}) + \dots + $0.01 (2^{29}),$

where a1=0.01, r=2 $a_{1} = 0.01, r = 2$ , and n=30 $n = 30$ . Using the formula

S n = a 1 ( 1 - r n ) 1 - r,

$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r},$

we have

S 30 = $0.01 ( 1 - 2 30 ) 1 - 2 = $10, 737, 418.23.

$S_{30} = \frac{$0.01 (1 - 2^{30})}{1 - 2} = $10, 737, 418.23.$

The pay exceeds $10.7 million for the month.

Now Try Exercise 57.

Example 9

The Amount of an Annuity. An annuity is a sequence of equal payments, made at equal time intervals, that earn interest. Fixed deposits in a savings account are an example of an annuity. Suppose that to save money to buy a car, Jacob deposits $2000 at the end of each of 5 years in an account that pays 3% interest, compounded annually. The total amount in the account at the end of 5 years is called the amount of the annuity. Find that amount.

Solution

The following time diagram can help visualize the problem. Note that no deposit is made until the end of the first year.

The amount of the annuity is the geometric series

$2000 + $2000 (1.03) 1 + $2000 (1.03) 2 + $2000 (1.03) 3 + $2000 (1.03) 4,

$$2000 + $2000 {(1.03)}^{1} + $2000 {(1.03)}^{2} + $2000 {(1.03)}^{3} + $2000 {(1.03)}^{4},$

where a1=$2000, n=5 $a_{1} = $2000, n = 5$ , and r=1.03 $r = 1.03$ . Using the formula

S n = a 1 ( 1 - r n ) 1 - r,

$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r},$

we have

S 5 = $2000 ( 1 - 1.03 5 ) 1 - 1.03 \approx $10, 618.27.

$S_{5} = \frac{$2000 (1 - {1.03}^{5})}{1 - 1.03} \approx $10, 618.27.$

The amount of the annuity is $10,618.27.

Now Try Exercise 61.

Example 10

The Economic Multiplier. Large sporting events have a significant impact on the economy of the host city. Super Bowl XLVII, hosted by New Orleans, generated a $480-million net impact for the region (Source: NewOrleansSaints.com, posted April 18, 2013, Marius M. Mihai, Research Analyst of the Division of Business and Economic Research at the University of New Orleans (DBER)). Assume that 60% of that amount is spent again in the area, and then 60% of that amount is spent again, and so on. This is known as the economic multiplier effect. Find the total effect on the economy.

Solution

The total economic effect is given by the infinite series

$480, 000, 000 + $480, 000, 000 (0.6) + $480, 000, 000 (0.6) 2 + \dots .

$$480, 000, 000 + $480, 000, 000 (0.6) + $480, 000, 000 {(0.6)}^{2} + \dots .$

Since |r|=|0.6|=0.6<1 $| r | = | 0.6 | = 0.6 < 1$ , the series has a sum. Using the formula for the sum of an infinite geometric series, we have

S \infty = a 1 1 - r = $480 , 000 , 000 1 - 0.6 = $1, 200, 000, 000.

$S_{\infty} = \frac{a_{1}}{1 - r} = \frac{$480, 000, 000}{1 - 0.6} = $1, 200, 000, 000.$

The total effect of the spending on the economy is $1,200,000,000.

Now Try Exercise 67.

Visualizing the Graph

Match the equation with its graph.

${(x - 1)}^{2} + {(y + 2)}^{2} = 9$
$y = x^{3} - x^{2} + x - 1$
$f (x) = 3^{x}$
$f (x) = x$
$a_{n} = n$
$y = \log (x + 3)$
$f (x) = - {(x - 2)}^{2} + 1$
$f (x) = {(x - 2)}^{2} - 1$
$y = \frac{1}{x - 1}$
$y = - 3 x + 4$

Answers on page A-42

11.3 Exercise Set

Find the common ratio.

1. 2, 4, 8, 16, …
2. 18, −6, 2, $- \frac{2}{3}, \dots$
3. −1, 1, −1, 1, …
4. −8, −0.8, −0.08, −0.008,…
5. $\frac{2}{3}, - \frac{4}{3}, \frac{8}{3}, - \frac{16}{3}, \dots$
6. 75, 15, 3, $\frac{3}{5}, \dots$
7. 6.275, 0.6275, 0.06275,…
11. $\frac{1}{x}, \frac{1}{x^{2}}, \frac{1}{x^{3}}, \dots$
9. $5, \frac{5 a}{2}, \frac{5 a^{2}}{4}, \frac{5 a^{3}}{8}, \dots$
10. $780, $858, $943.80, $1038.18,…

Find the indicated term.

11. 2, 4, 8, 16, …; the 7th term
12. 2, −10, 50, −250,…; the 9th term
13. 2, $2 \sqrt{3}$ , 6, …; the 9th term
14. 1, −1, 1, −1,…; the 57th term
15. $\frac{7}{625}, - \frac{7}{25}, \dots;$ the 23rd term
16. $1000, $1060, $1123.60, …; the 5th term

Find the nth, or general, term.

17. 1, 3, 9, …
18. 25, 5, 1,…
19. 1, −1, 1, −1,…
20. −2, 4, −8,…
21. $\frac{1}{x}, \frac{1}{x^{2}}, \frac{1}{x^{3}}, \dots$
22. $5, \frac{5 a}{2}, \frac{5 a^{2}}{4}, \frac{5 a^{3}}{8}, \dots$
23. Find the sum of the first 7 terms of the geometric series

$6 + 12 + 24 + \dots .$
24. Find the sum of the first 10 terms of the geometric series

$16 - 8 + 4 - \dots .$
25. Find the sum of the first 9 terms of the geometric series

$\frac{1}{18} - \frac{1}{6} + \frac{1}{2} - \dots .$
26. Find the sum of the geometric series

$- 8 + 4 + (- 2) + \dots + (- \frac{1}{32}) .$

Determine whether the statement is true or false.

27. The sequence $2, - 2 \sqrt{2}, 4, - 4 \sqrt{2}, 8, \dots$ is geometric.
28. The sequence with general term 3n is geometric.
29. The sequence with general term $2^{n}$ is geometric.
30. Multiplying a term of a geometric sequence by the common ratio produces the next term of the sequence.
31. An infinite geometric series with common ratio −0.75 has a sum.
32. Every infinite geometric series has a limit.

Find the sum, if it exists.

33. $4 + 2 + 1 + \dots$
34. $7 + 3 + \frac{9}{7} + \dots$
35. $25 + 20 + 16 + \dots$
36. $100 - 10 + 1 - \frac{1}{10} + \dots$
37. $8 + 40 + 200 + \dots$
38. $- 6 + 3 - \frac{3}{2} + \frac{3}{4} - \dots$
39. $0.6 + 0.06 + 0.006 + \dots$
40. $\sum_{k = 0}^{10} 3^{k}$
41. $\sum_{k = 1}^{11} 15 {(\frac{2}{3})}^{k}$
42. $\sum_{k = 0}^{50} 200 {(1.08)}^{k}$
43. $\sum_{k = 1}^{\infty} {(\frac{1}{2})}^{k - 1}$
44. $\sum_{k = 1}^{\infty} 2^{k}$
45. $\sum_{k = 1}^{\infty} {12.5}^{k}$
46. $\sum_{k = 1}^{\infty} 400 {(1.0625)}^{k}$
47. $\sum_{k = 1}^{\infty} $500 {(1.11)}^{- k}$
48. $\sum_{k = 1}^{\infty} $1000 {(1.06)}^{- k}$
49. $\sum_{k = 1}^{\infty} 16 {(0.1)}^{k - 1}$
50. $\sum_{k = 1}^{\infty} \frac{8}{3} {(\frac{1}{2})}^{k - 1}$

Find fraction notation.

51. 0.131313…, or $0. \bar{13}$
52. 0.2222…, or $0. \bar{2}$
53. $8.999 \bar{9}$
54. $6.1616 \bar{16}$
55. $3.4125 \bar{125}$
56. $12.7809 \bar{809}$
57. Daily Doubling Salary. Suppose that someone offered you a job for the month of February (28 days) under the following conditions. You will be paid $0.01 the 1st day, $0.02 the 2nd, $0.04 the 3rd, and so on, doubling your previous day’s salary each day. How much would you earn altogether for the month?
58. Bouncing Ping-Pong Ball. A ping-pong ball is dropped from a height of 16 ft and always rebounds $\frac{1}{4}$ of the distance fallen.
1. How high does it rebound the 6th time?
2. Find the total sum of the rebound heights of the ball.
59. Bungee Jumping. A bungee jumper always rebounds 60% of the distance fallen. A bungee jump is made using a cord that stretches to 200 ft.
1. After jumping and then rebounding 9 times, how far has a bungee jumper traveled upward (the total rebound distance)?
2. About how far will a jumper have traveled upward (bounced) before coming to rest?
60. Population Growth. A coastal town has a present population of 32,100, and the population is increasing by 3% each year.
1. What will the population be in 15 years?
2. How long will it take for the population to double?
61. Amount of an Annuity. To save for the down payment on a house, the Clines make a sequence of 10 yearly deposits of $3200 each in a savings account on which interest is compounded annually at 4.6%. Find the amount of the annuity.
62. Amount of an Annuity. To create a college fund, a parent makes a sequence of 18 yearly deposits of $1000 each in a savings account on which interest is compounded annually at 3.2%. Find the amount of the annuity.
63. Doubling the Thickness of Paper. A piece of paper is 0.01 in. thick. It is cut and stacked repeatedly in such a way that its thickness is doubled each time for 20 times. How thick is the result?
64. Amount of an Annuity. A sequence of yearly payments of P dollars is invested at the end of each of N years at interest rate i, compounded annually. The total amount in the account, or the amount of the annuity, is V.
1. Show that
  
  $V = \frac{P [{(1 + i)}^{N} - 1]}{i} .$
2. Suppose that interest is compounded n times per year and deposits are made every compounding period. Show that the formula for V is then given by
  
  $V = \frac{P [{(1 + \frac{i}{n})}^{n N} - 1]}{i / n} .$
65. Amount of an Annuity. A sequence of payments of $300 is invested over 12 years at the end of each quarter at 5.1%, compounded quarterly. Find the amount of the annuity. Use the formula in Exercise 64(b).
66. Amount of an Annuity. A sequence of yearly payments of $750 is invested at the end of each of 10 years at 4.75%, compounded annually. Find the amount of the annuity. Use the formula in Exercise 64(a).
67. The Economic Multiplier. Suppose that the government is making a $13,000,000,000 expenditure to stimulate the economy. If 85% of this is spent again, and so on, what is the total effect on the economy?
68. Advertising Effect. Gigi’s Cupcake Truck is about to open for business in a city of 3,000,000 people, traveling to several curbside locations in the city each day to sell cupcakes. The owners plan an advertising campaign that they think will induce 30% of the people to buy their cupcakes. They estimate that if those people like the product, they will induce $30 % \cdot 30 % \cdot 3, 000, 000$ more to buy the product, and those will induce $30 % \cdot 30 % \cdot 30 % \cdot 3, 000, 000$ and so on. In all, how many people will buy Gigi’s cupcakes as a result of the advertising campaign? What percentage of the population is this?

Skill Maintenance

For each pair of functions, find $(f \circ g) (x)$ and $(g \circ f) (x)$ . [2.3]

69. $f (x) = x^{2}, g (x) = 4 x + 5$
70. $f (x) = x - 1, g (x) = x^{2} + x + 3$

Solve. [5.5]

71. $5^{x} = 35$
72. $\log_{2} x = - 4$

Synthesis

73. Prove that $\sqrt{3} - \sqrt{2}, 4 - \sqrt{6}$ , and $6 \sqrt{3} - 2 \sqrt{2}$ form a geometric sequence.
74. Assume that $a_{1}, a_{2}, a_{3}, \dots$ is a geometric sequence. Prove that $\ln a_{1}, {ln a}_{2}, {ln a}_{3}, \dots$ is an arithmetic sequence.
75. Consider the sequence

$x + 3, x + 7, 4 x - 2, \dots .$
1. If the sequence is arithmetic, find x and then determine each of the 3 terms and the 4th term.
2. If the sequence is geometric, find x and then determine each of the 3 terms and the 4th term.
76. Find the sum of the first n terms of

$1 + x + x^{2} + \dots .$
77. Find the sum of the first n terms of

$x^{2} - x^{3} + x^{4} - x^{5} + \dots .$
78. The sides of a square are 16 cm long. A second square is inscribed by joining the midpoints of the sides, successively. In the second square, we repeat the process, inscribing a third square. If this process is continued indefinitely, what is the sum of all the areas of all the squares? (Hint: Use an infinite geometric series.)

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Table of Contents for 11.3 Geometric Sequences and Series

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Table of Contents for
11.3 Geometric Sequences and Series