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Mathematics LibreTexts

12.4E: Exercises

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Practice Makes Perfect

Exercise 12.4E.21 Determine if a Sequence is Geometric

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

  1. 3,12,48,192,768,3072,
  2. 2,10,50,250,1250,6250,
  3. 48,24,12,6,3,32,
  4. 54,18,6,2,23,29,
  5. 3,6,12,24,48,96,
  6. 2,6,18,54,162,486,
Answer

1. The sequence is geometric with common ratio r=4.

3. The sequence is geometric with common ratio r=12.

5. The sequence is geometric with common ratio r=2.

Exercise 12.4E.22 Determine if a Sequence is Geometric

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

  1. 48,24,12,6,3,32,
  2. 12,6,0,6,12,18,
  3. 7,2,3,8,13,18,
  4. 5,9,13,17,21,25,
  5. 12,14,18,116,132,164,
  6. 4,8,12,24,48,96,
Answer

1. The sequence is geometric with common ratio r=12.

3. The sequence is arithmetic with common difference d=5.

5. The sequence is geometric with common ratio r=12.

Exercise 12.4E.23 Determine if a Sequence is Geometric

In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.

  1. a1=4 and r=3
  2. a1=9 and r=2
  3. a1=4 and r=2
  4. a1=5 and r=3
  5. a1=27 and r=13
  6. a1=64 and r=14
Answer

1. 4,12,36,108,324

3. 4,8,16,32,64

5. 27,9,3,1,13

Exercise 12.4E.24 Find the General Term (nth Term) of a Geometric Sequence
  1. Find a11 given a1=8 and r=3.
  2. Find a13 given a1=7 and r=2.
  3. Find a10 given a1=6 and r=2.
  4. Find a15 given a1=4 and r=3.
  5. Find a10 given a1=100,000 and r=0.1.
  6. Find a8 given a1=1,000,000 and r=0.01.
Answer

1. 472,392

3. 3,072

5. 0.0001

Exercise 12.4E.25 Find the General Term (nth Term) of a Geometric Sequence

In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.

  1. Find a9 of the sequence, 9,18,36,72,144,288,
  2. Find a12 of the sequence, 5,15,45,135,405,1215,
  3. Find a15 of the sequence, 486,162,54,18,6,2,
  4. Find a16 of the sequence, 224,112,56,28,14,7,
  5. Find a10 of the sequence, 1,0.1,0.01,0.001,0.0001,0.00001,
  6. Find a9 of the sequence, 1000,100,10,1,0.1,0.01,
Answer

1. a9=2,304. The general term is an=9(2)n1.

3. a15=219,683. The general term is an=486(13)n1.

5. a10=0.000000001. The general term is an=(0.1)n1.

Exercise 12.4E.26 Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

  1. 8,24,72,216,648,1944,
  2. 7,14,28,56,112,224,
  3. 6,12,24,48,96,192,
  4. 4,12,36,108,324,972,
  5. 81,27,9,3,1,13,
  6. 256,64,16,4,1,14,116,
Answer

1. 57,395,624

3. 65,538

5. 7,174,45359,049121.5

Exercise 12.4E.27 Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the geometric sequence.

  1. 15i=1(2)i
  2. 10i=1(3)i
  3. 9i=14(2)i
  4. 8i=15(3)i
  5. 10i=19(13)i
  6. 15i=14(12)i
Answer

1. 65,534

3. 4088

5. 29,52465614.5

Exercise 12.4E.28 Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

  1. 1+13+19+127+181+1243+1729+
  2. 1+12+14+18+116+132+164+
  3. 62+2329+227281+
  4. 4+21+1214+18
  5. 6+12+24+48+96+192+
  6. 5+15+45+135+405+1215+
  7. 1,024+512+256+128+64+32+
  8. 6,561+2187+729+243+81+27+
Answer

1. 32

3. 92

5. no sum as r1

7. \(2,048\(

Exercise 12.4E.29 Find the Sum of an Infinite Geometric Series

In the following exercises, write each repeating decimal as a fraction.

  1. 0.¯3
  2. 0.¯6
  3. 0.¯7
  4. 0.¯2
  5. 0.¯45
  6. 0.¯27
Answer

1. 13

3. 79

5. 511

Exercise 12.4E.30 Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

  1. Find the total effect on the economy of each government tax rebate to each household in order to stimulate the economy if each household will spend the indicated percent of the rebate in goods and services.
Tax rebate to each household Percent spent on goods and services Total Effect on the economy
a. $1,000 85%  
b. $1,000 75%  
c. $1,500 90%  
d. $1,500 80%  
Table 12.3.3

2. New grandparents decide to invest $100 per month in an annuity for their grandchild. The account will pay 6% interest per year which is compounded monthly (12 times a year). How much will be in the child’s account at their twenty-first birthday?

3. Berenice just got her first full-time job after graduating from college at age 30. She decided to invest $500 per quarter in an IRA (an annuity). The interest on the annuity is 7% which is compounded quarterly (4 times a year). How much will be in the Berenice’s account when she retires at age 65?

4. Alice wants to purchase a home in about five years. She is depositing $500 a month into an annuity that earns 5% per year that is compounded monthly (12 times a year). How much will Alice have for her down payment in five years?

5. Myra just got her first full-time job after graduating from college. She plans to get a master’s degree, and so is depositing $2,500 a year from her year-end bonus into an annuity. The annuity pays 6.5% per year and is compounded yearly. How much will she have saved in five years to pursue her master’s degree?

Answer

1. a. $6666.67 b. $4000 c. $15,000 d. $7500

3. $295,581.88

5. $14,234.10

Exercise 12.4E.31 Writing Exercises
  1. In your own words, explain how to determine whether a sequence is geometric.
  2. In your own words, explain how to find the general term of a geometric sequence.
  3. In your own words, explain the difference between a geometric sequence and a geometric series.
  4. In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.
Answer

2. Answers will vary.

4. Answers will vary

Self Check

This figure shows seven rows and four columns. The first row is the header row and reads, “I can”, “Confidently”, “With some help”, and “No, I don’t get it. The first column reads, “Determine if a sequence is geometric”, “Find the general term (nth term) of a”, “Geometric sequence”, “Find the sum of an Infinite geometric series”, Use geometric sequences to solve applications”. The remaining columns are blank.
Figure 12.3.11
  1. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
  2. What does this checklist tell you about your mastery of this section? What steps will you take to improve?

This page titled 12.4E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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