Determine Whether Each Sequence Is Arithmetic, Geometric, Or Neither

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7-7 Geometric Sequences as Exponential Functions

Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1. 200, 40, 8, … SOLUTION:

Since the ratios are constant, the sequence is geometric. The common ratio is .

2. 2, 4, 16, … SOLUTION:

The ratios are not constant, so the sequence is not geometric.

There is no common difference, so the sequence is not arithmetic.

Thus, the sequence is neither geometric nor arithmetic.

3. −6, −3, 0, 3, … SOLUTION:

The ratios are not constant, so the sequence is not geometric.

Since the differences are constant, the sequence is arithmetic. The common difference is 3.

4. 1, −1, 1, −1, … SOLUTION:

Since the ratios are constant, the sequence is geometric. The common ratio is –1.

Find the next three terms in each geometric sequence. 5. 10, 20, 40, 80, … SOLUTION: eSolutions Manual - Powered by Cognero Page 1

The common ratio is 2. Multiply each term by the common ratio to find the next three terms. 80 × 2 = 160 160 × 2 = 320 320 × 2 = 640 The next three terms of the sequence are 160, 320, and 640.

6. 100, 50, 25, … SOLUTION: Calculate common ratio.

The common ratio is 0.5. Multiply each term by the common ratio to find the next three terms. 25 × 0.5 = 12.5 12.5 × 0.5 = 6.25 6.25 × 0.5 = 3.125 The next three terms of the sequence are 12.5, 6.25, and 3.125.

7. 4, −1, , …

SOLUTION: Calculate the common ratio.

The common ratio is . Multiply each term by the common ratio to find the next three terms.

× =

The next three terms of the sequence are , , and .

8. −7, 21, −63, … SOLUTION: Calculate the common ratio.

The common ratio is –3. Multiply each term by the common ratio to find the next three terms. –63 × –3 = 189 189 × –3 = –567 –567 × –3 = 1701 The next three terms of the sequence are 189, −567, and 1701.

Write an equation for the nth term of the geometric sequence, and find the indicated term. 9. Find the fifth term of −6, −24, −96, … SOLUTION: Calculate the common ratio.

n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is 4, n−1 so r = 4. The first term is –6, so a1 = –6. Then, an = −6 • (4) .

The 5th term of the sequence is –1536.

10. Find the seventh term of −1, 5, −25, … SOLUTION: Calculate the common ratio.

n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is –5, n−1 so r = –5. The first term is –1, so a1 = –1. Then, an = −1 • (–5) .

The 7th term of the sequence is –15,625.

11. Find the tenth term of 72, 48, 32, … SOLUTION: Calculate the common ratio.

n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is ,

so r = . The first term is 72, so a1 = 72. Then, an = 72 • .

The 10th term of the sequence is .

12. Find the ninth term of 112, 84, 63, … SOLUTION: Calculate the common ratio.

n – 1 Use the formula an = a1r to write an equation for the nth term of the geometric series. The common ratio is ,

so r = . The first term is 112, so a1 = 112. Then, an = 112 • .

The 9th term of the sequence is .

13. EXPERIMENT In a physics class experiment, Diana drops a ball from a height of 16 feet. Each bounce has 70% the height of the previous bounce. Draw a graph to represent the height of the ball after each bounce. SOLUTION: Make a table of values. Bounce Ball Height 1 0.7(16) = 11.2 2 0.7(11.2) = 7.84 3 0.7(7.84) = 5.488 4 0.7(5.488) = 3.8416 5 0.7(3.8416) = 2.68912 6 0.7(2.68912) = 1.882384 7 0.7(1.882384) =1.3176688 Graph the bounce on the x-axis and the ball height on the y-axis.

Determine whether each sequence is arithmetic, geometric, or neither. Explain. 14. 4, 1, 2, … SOLUTION: Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the ratios of the differences of consecutive terms

There is no common difference, so the sequence is not arithmetic.

Thus, the sequence is neither geometric nor arithmetic.

15. 10, 20, 30, 40 … SOLUTION: Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the differences of consecutive terms.

Since the differences are constant, the sequence is arithmetic. The common difference is 10.

16. 4, 20, 100, … SOLUTION: Find the ratios of consecutive terms.

Since the ratios are constant, the sequence is geometric. The common ratio is 5.

17. 212, 106, 53, … SOLUTION: Find the ratios of consecutive terms.

Since the ratios are constant, the sequence is geometric. The common ratio is .

18. −10, −8, −6, −4 … SOLUTION: Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the differences of consecutive terms.

Since the differences are constant, the sequence is arithmetic. The common difference is 2.

19. 5, −10, 20, 40, … SOLUTION: Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the differences of consecutive terms.

There is no common difference, so the sequence is not arithmetic. Thus, the sequence is neither geometric nor arithmetic.

Find the next three terms in each geometric sequence. 20. 2, −10, 50, … SOLUTION: Calculate the common ratio.

The common ratio is –5. Multiply each term by the common ratio to find the next three terms. 50 × –5 = –250 –250 × –5 = 1250 1250 × –5 = –6250 The next three terms of the sequence are −250, 1250, and −6250.

21. 36, 12, 4, … SOLUTION: Calculate the common ratio.

The common ratio is . Multiply each term by the common ratio to find the next three terms. 4 × = × = × = The next three terms of the sequence are , , and .

22. 4, 12, 36, … SOLUTION: Calculate the common ratio.

The common ratio is 3. Multiply each term by the common ratio to find the next three terms. 36 × 3 = 108 108 × 3 = 324 324 × 3 = 972 The next three terms of the sequence are 108, 324, and 972.

23. 400, 100, 25, … SOLUTION: Calculate the common ratio.

The common ratio is . Multiply each term by the common ratio to find the next three terms. 25 × = × = × = The next three terms of the sequence are , , and .

24. −6, −42, −294, … SOLUTION: Calculate the common ratio.

The common ratio is 7. Multiply each term by the common ratio to find the next three terms. –294 × 7 = –2058 –2058 × 7 = –14,406 –14,406 × 7 = –100,842 The next three terms of the sequence are −2058, −14,406, and −100,842.

25. 1024, −128, 16, … SOLUTION: Calculate the common ratio.

The common ratio is . Multiply each term by the common ratio to find the next three terms.

16 × = –2

–2 × =

× =

The next three terms of the sequence are −2, , and .

26. The first term of a geometric series is 1 and the common ratio is 9. What is the 8th term of the sequence? SOLUTION:

The 8th term of the sequence is 4,782,969.

27. The first term of a geometric series is 2 and the common ratio is 4. What is the 14th term of the sequence? SOLUTION:

The 14th term of the sequence is 134,217,728.

28. What is the 15th term of the geometric sequence −9, 27, −81, …? SOLUTION: Calculate the common ratio.

The common ratio is –3.

The 15th term of the sequence is –43,046,721.

29. What is the 10th term of the geometric sequence 6, −24, 96, …? SOLUTION: Calculate the common ratio.

The common ratio is –4.

The 10th term of the sequence is –1,572,864.

30. PENDULUM A pendulum swings with an arc length of 24 feet on its first swing. On each swing after the first swing, the arc length is 60% of the length of the previous swing. Draw a graph that represents the arc length after each swing. SOLUTION: Make a table of values. Swing Arc Length 1 24 2 0.6(24) = 14.4 3 0.6(14.4) = 8.64 4 0.6(8.64) = 5.184 5 0.6(5.184) = 3.1104 6 0.6(3.1104) = 1.86624 Graph the swing on the x-axis and the arc length on the y-axis.

31. Find the eighth term of a geometric sequence for which a3 = 81 and r = 3. SOLUTION:

Because a3 = 81, the third term in the sequence is 81. To find the eighth term of the sequence, you need to find the 1st term of the sequence. Use the nth term of a Geometric Sequence formula.

Then a1 is 9.

Use a1 to find the eighth term of the sequence.

The eighth term of the geometric sequence is 19,683.

32. CCSS REASONING At an online mapping site, Mr. Mosley notices that when he clicks a spot on the map, the map zooms in on that spot. The magnification increases by 20% each time. a. Write a formula for the nth term of the geometric sequence that represents the magnification of each zoom level. (Hint: The common ratio is not just 0.2.) b. What is the fourth term of this sequence? What does it represent? SOLUTION: a. Because the magnification increases by 20% with each click, the total magnification after each click is 120%. The common ratio is 1.2. To find the nth term of the geometric sequence that represents the magnification of each zoom n level, use the formula 1.2 . b.

The fourth term of the sequence is 2.0736. It represents the magnification after the fourth click. So, the map will be magnified at approximately 207% of the original size after the fourth click.

33. ALLOWANCE Danielle’s parents have offered her two different options to earn her allowance for a 9-week period over the summer. She can either get paid $30 each week or $1 the first week, $2 for the second week, $4 for the third week, and so on. a. Does the second option form a geometric sequence? Explain. b. Which option should Danielle choose? Explain. SOLUTION: a. Calculate the common ratio.

There is a common ratio of 2. So, the second option does form a geometric sequence. b. Calculate how much Danielle would earn with each option. Option 1 9(30) = 270 Option 2 1 + 2 + 4 + 8+ 16 + 32 + 64 + 128 + 256 or 511 In nine weeks, Danielle would earn $270 with the first option and $511 with the second option. So, she should choose the second option.

34. SIERPINSKI’S TRIANGLE Consider the inscribed equilateral triangles shown. The perimeter of each triangle is one half of the perimeter of the next larger triangle. What is the perimeter of the smallest triangle?

SOLUTION:

This is a geometric sequence. The first term is 3(40) or 120 and the common ratio is . To find the perimeter of the smallest triangle, find the 5th term of the sequence.

The perimeter of the smallest triangle is 7.5 centimeters.

35. If the second term of a geometric sequence is 3 and the third term is 1, find the first and fourth terms of the sequence. SOLUTION: Divide the 3rd term by the 2nd term to find the common ratio. The common ratio is . Substitute 2 for n and for r to find the first term.

The first term is 9. Find the 4th term.

The fourth term is .

36. If the third term of a geometric sequence is −12 and the fourth term is 24, find the first and fifth terms of the sequence. SOLUTION: Divide the 4th term by the 3rd term to find the common ratio. The common ratio is or –2. Substitute 3 for n and –2 for r to find the first term.

The first term is –3. Find the fifth term.

The fifth term is 48.

37. EARTHQUAKES The Richter scale is used to measure the force of an earthquake. The table shows the increase in magnitude for the values on the Richter scale. Richter Increase in Rate of Number Magnitude Change (x) (y) (slope) 1 1 − 2 10 9 3 100 4 1000 5 10,000 a. Copy and complete the table. Remember that the rate of change is the change in y divided by the change in x. b. Plot the ordered pairs (Richter number, increase in magnitude). c. Describe the graph that you made of the Richter scale data. Is the rate of change between any two points the same? d. Write an exponential equation that represents the Richter scale. SOLUTION: a. Richter Increase in Rate of Change (slope) Number (x) Magnitude (y) 1 1 − 2 10 9 3 100

4 1000

5 10,000

b. Graph the Richter number on the x-axis and the increase in magnitude on the y-axis.

c. The graph appears to be exponential. The rate of change between any two points does not match any others. d. Calculate the common ratio.

There is a common ratio of 10, so this is situation can be modeled by a geometric sequence. The exponential equation that represents the Richter scale is y = 1 • (10)x−1.

38. CHALLENGE Write a sequence that is both geometric and arithmetic. Explain your answer. SOLUTION: The sequence 1, 1, 1, 1, … is both geometric and arithmetic. The common ratio is 1 making it a geometric sequence. The common difference is 0, making it an arithmetic sequence as well.

This can be done for any value n. n, n, n, ... is arithmetic and geometric.

39. CCSS CRITIQUE Haro and Matthew are finding the ninth term of the geometric sequence −5, 10, −20, … . Is either of them correct? Explain your reasoning.

SOLUTION: The common ratio of the sequence is –2.

The ninth term of the sequence is –1280. Neither Haro nor Matthew is correct. Haro calculated the exponent incorrectly. Matthew did not enclose the common ratio in parentheses which caused him to make a sign error.

40. REASONING Write a sequence of numbers that form a pattern but are neither arithmetic nor geometric. Explain the pattern. SOLUTION: The sequence 1, 4, 9, 16, 25, 36, … has a pattern, because each number is a perfect square. However, there is no common ratio which means it is not a geometric sequence. There is no common difference, which means it is not an arithmetic sequence.

41. WRITING IN MATH How are graphs of geometric sequences and exponential functions similar? different? SOLUTION: Sample answer: When graphed, the terms of a geometric sequence lie on a curve that can be represented by an exponential function. They are different in that the domain of a geometric sequence is the set of natural numbers, while the domain of an exponential function is all real numbers. Thus, geometric sequences are discrete, while exponential functions are continuous.

n–1 For example, the geometric sequence 1, 2, 4, 8, ... has a = 1, r = 2, and the nth term given by an = 1(2) , n–1 where n is any positive integer. The graph of the function an = 1(2) would be as follows.

The exponential function given by y = 1(2)x–1will generate similar values but the domain of x is all real numbers. The graph of this function is below.

Even though, the two graphs contain many of the same points, the graph of the geometric sequence is discrete while the graph of the exponential function is continuous.

43. Find the eleventh term of the sequence 3, −6, 12, −24, … A 1024 B 3072 C 33 D −6144 SOLUTION: Calculate the common ratio.

The common ratio is –2.

The eleventh term of the sequence is 3072. Choice B is the correct answer.

44. What is the total amount of the investment shown in the table below if interest is compounded monthly?

F $613.56 G $616.00 H $616.56 J $718.75 SOLUTION: Use the equation for compound interest, with P = 500, r = 0.0525, n = 12, and t = 4.

The total amount of the investment is about $616.56. Choice H is the correct answer.

45. SHORT RESPONSE Gloria has $6.50 in quarters and dimes. If she has 35 coins in total, how many of each coin does she have? SOLUTION: Let q = the number of quarters and let d = the number of dimes. Then, q + d = 35 and 0.25q + 0.10d = 6.50. Solve the first equation for d.

Substitute 35 – q for d in the second equation and solve for q.

Use the value of q and either equation to find the value of d.

Gloria has 15 dimes and 20 quarters.

x 46. What are the domain and range of the function y = 4(3 ) – 2?

A D = {all real numbers}, R = {y | y > –2} B D = {all real numbers}, R = {y | y > 0} C D = {all integers}, R = {y | y > –2} D D = {all integers}, R = {y | y > 0} SOLUTION: Use a graphing calculator to graph the function Y1= 4(3x) – 2.

The graph is continuous from left to right and increases from –2 to infinity.Thus, the domain is all real numbers and the range is all real numbers greater than –2. Therefore, the correct choice is A.

Find the next three terms in each geometric sequence. 47. 2, 6, 18, 54, … SOLUTION: Calculate the common ratio.

The common ratio is 3. Multiply each term by the common ratio to find the next three terms. 54 × 3 = 162 162 × 3 = 486 486 × 3 = 1458 The next three terms of the sequence are 162, 486, and 1458

48. −5, −10, −20, −40, … SOLUTION: Calculate the common ratio.

The common ratio is 2. Multiply each term by the common ratio to find the next three terms. –40 × 2 = –80 –80 × 2 = –160 –160 × 2 = –320 The next three terms of the sequence are −80, −160, and −320.

49.

SOLUTION: Calculate the common ratio.

The common ratio is . Multiply each term by the common ratio to find the next three terms.

× =

The next three terms of the sequence are , , and .

50. −3, 1.5, −0.75, 0.375, … SOLUTION: Calculate the common ratio.

The common ratio is –0.5. Multiply each term by the common ratio to find the next three terms. 0.375 × –0.5 = –0.1875 –0.1875 × –0.5 = 0.09375 0.09375 × –0.5 = –0.046875 The next three terms of the sequence are −0.1875, 0.09375, and −0.046875.

51. 1, 0.6, 0.36, 0.216, … SOLUTION: Calculate the common ratio.

The common ratio is 0.6. Multiply each term by the common ratio to find the next three terms. 0.216 × 0.6 = 0.1296 0.1296 × 0.6 = 0.7776 0.7776 × 0.6 = 0.046656 The next three terms of the sequence are 0.1296, 0.07776, and 0.046656.

52. 4, 6, 9, 13.5, … SOLUTION: Calculate the common ratio.

The common ratio is 1.5. Multiply each term by the common ratio to find the next three terms. 13.5 × 1.5 = 20.25 20.25 × 1.5 = 30.375 30.375 × 1.5 = 45.5625 The next three terms of the sequence are 20.25, 30.375, and 45.5625.

Graph each function. Find the y-intercept and state the domain and range.

53.

SOLUTION: x y

–2 11

–1 –1

0 –4

The function crosses the y-axis at –4. The domain is all real numbers, and the range is all real numbers greater than –5.

54. y = 2(4)x SOLUTION: x x (4) y –2 –2 (4) =

–1 –1 (4) =

0 (4)0 = 1 2 1 1 (4) = 4 8 2 (4) 2 = 16 32

The function crosses the y-axis at 2. The domain is all real numbers, and the range is all real numbers greater than 0.

55.

SOLUTION: x x (3) y –2 –2 (3) =

–1 –1 (3) =

0 (3)0 = 1

1 (3)1 = 3

2 (3) 2 = 9

The function crosses the y-axis at . The domain is all real numbers, and the range is all real numbers greater than 0.

56. LANDSCAPING A blue spruce grows an average of 6 inches per year. A hemlock grows an average of 4 inches per year. If a blue spruce is 4 feet tall and a hemlock is 6 feet tall, write a system of equations to represent their growth. Find and interpret the solution in the context of the situation. SOLUTION: Let x = the number of years the tree grows and let y = the height of the tree after x years. So, y = 48 + 6x, and y = 72 + 4x. Substitute 48 + 6x for y in the second equation.

Use the value for x and either equation to find the value for y.

The solution is (12, 120). This means that in 12 years the trees will be the same height, 120 inches or 10 feet.

57. MONEY City Bank requires a minimum balance of $1500 to maintain free checking services. If Mr. Hayashi is going to write checks for the amounts listed in the table, how much money should he start with in order to have free checking?

SOLUTION: Let x = the amount Mr. Hayashi should start with in order to have free checking. Then, x – (1300 + 947) ≥ 1500.

Mr. Hayashi should start with at least $3747.

Write an equation in slope-intercept form of the line with the given slope and y-intercept. 58. slope: 4, y-intercept: 2 SOLUTION: The slope-intercept form of a line is y = mx + b, where m = the slope and b = the y-intercept. So, y = 4x + 2.

59. slope: −3, y-intercept:

SOLUTION:

The slope-intercept form of a line is y = mx + b, where m = the slope and b = the y-intercept. So, .

60. slope: , y-intercept: −5

SOLUTION:

The slope-intercept form of a line is y = mx + b, where m = the slope and b = the y-intercept. So, .

61. slope: , y-intercept: −9

SOLUTION:

The slope-intercept form of a line is y = mx + b, where m = the slope and b = the y-intercept. So, .

62. slope: , y-intercept:

SOLUTION:

The slope-intercept form of a line is y = mx + b, where m = the slope and b = the y-intercept. So, .

63. slope: −6, y-intercept: −7 SOLUTION: The slope-intercept form of a line is y = mx + b, where m = the slope and b = the y-intercept. So, y = −6x − 7.

Simplify each expression. If not possible, write simplified. 64. 3u + 10u SOLUTION: Since 3u and 10u are like terms, this expression can be simplified.

65. 5a – 2 + 6a SOLUTION: Since 5a and 6a are like terms, the expression can be simplified.

66. 6m2 – 8m SOLUTION: 2 6m and 8m are not like terms. Therefore, this expression is already simplified.

67. 4w2 + w + 15w2 SOLUTION: 2 2 Since 4w and 15w are like terms, this expression can be simplified.

68. 13(5 + 4a) SOLUTION: Since this expression has indicated multiplication, the Distributive Property can be used to simplify the expression.

69. (4t – 6)16 SOLUTION: Since this expression contains indicated multiplication, the Distributive Property can be used to simplify the expression.

Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1. 200, 40, 8, … SOLUTION:

Since the ratios are constant, the sequence is geometric. The common ratio is .

2. 2, 4, 16, … SOLUTION:

The ratios are not constant, so the sequence is not geometric.

There is no common difference, so the sequence is not arithmetic.

Thus, the sequence is neither geometric nor arithmetic.

3. −6, −3, 0, 3, … SOLUTION:

The ratios are not constant, so the sequence is not geometric.

Since the differences are constant, the sequence is arithmetic. The common difference is 3.

4. 1, −1, 1, −1, … SOLUTION:

7-7 Geometric Sequences as Exponential Functions Since the ratios are constant, the sequence is geometric. The common ratio is –1.

Find the next three terms in each geometric sequence. 5. 10, 20, 40, 80, … SOLUTION:

The common ratio is 2. Multiply each term by the common ratio to find the next three terms. 80 × 2 = 160 160 × 2 = 320 320 × 2 = 640 The next three terms of the sequence are 160, 320, and 640.

6. 100, 50, 25, … SOLUTION: Calculate common ratio.

7. 4, −1, , …

SOLUTION: Calculate the common ratio.

The common ratio is . Multiply each term by the common ratio to find the next three terms.

× =

The next three terms of the sequence are , , and .

8. −7, 21, −63, … eSolutions Manual - Powered by Cognero Page 2 SOLUTION: Calculate the common ratio.