14.4E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{21}\) Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
- \(3,12,48,192,768,3072, \dots\)
- \(2,10,50,250,1250,6250, \dots\)
- \(48,24,12,6,3, \frac{3}{2}, \dots\)
- \(54,18,6,2, \frac{2}{3}, \frac{2}{9}, \dots\)
- \(-3,6,-12,24,-48,96, \dots\)
- \(2,-6,18,-54,162,-486, \dots\)
- Answer
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1. The sequence is geometric with common ratio \(r=4\).
3. The sequence is geometric with common ratio \(r=\frac{1}{2}\).
5. The sequence is geometric with common ratio \(r=−2\).
Exercise \(\PageIndex{22}\) 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.
- \(48,24,12,6,3, \frac{3}{2}, \ldots\)
- \(12,6,0,-6,-12,-18, \dots\)
- \(-7,-2,3,8,13,18, \dots\)
- \(5,9,13,17,21,25, \ldots\)
- \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \dots\)
- \(4,8,12,24,48,96, \dots\)
- Answer
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1. The sequence is geometric with common ratio \(r=\frac{1}{2}\).
3. The sequence is arithmetic with common difference \(d=5\).
5. The sequence is geometric with common ratio \(r=\frac{1}{2}\).
Exercise \(\PageIndex{23}\) 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.
- \(a_{1}=4\) and \(r=3\)
- \(a_{1}=9\) and \(r=2\)
- \(a_{1}=-4\) and \(r=-2\)
- \(a_{1}=-5\) and \(r=-3\)
- \(a_{1}=27\) and \(r=\frac{1}{3}\)
- \(a_{1}=64\) and \(r=\frac{1}{4}\)
- Answer
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1. \(4,12,36,108,324\)
3. \(-4,8,-16,32,-64\)
5. \(27,9,3,1, \frac{1}{3}\)
Exercise \(\PageIndex{24}\) Find the General Term (\(n\)th Term) of a Geometric Sequence
- Find \(a_{11}\) given \(a_{1}=8\) and \(r=3\).
- Find \(a_{13}\) given \(a_{1}=7\) and \(r=2\).
- Find \(a_{10}\) given \(a_{1}=-6\) and \(r=-2\).
- Find \(a_{15}\) given \(a_{1}=-4\) and \(r=-3\).
- Find \(a_{10}\) given \(a_{1}=100,000\) and \(r=0.1\).
- Find \(a_{8}\) given \(a_{1}=1,000,000\) and \(r=0.01\).
- Answer
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1. \(472,392\)
3. \(3,072\)
5. \(0.0001\)
Exercise \(\PageIndex{25}\) Find the General Term (\(n\)th Term) of a Geometric Sequence
In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.
- Find \(a_{9}\) of the sequence, \(9,18,36,72,144,288, \dots\)
- Find \(a_{12}\) of the sequence, \(5,15,45,135,405,1215, \dots\)
- Find \(a_{15}\) of the sequence, \(-486,162,-54,18,-6,2, \dots\)
- Find \(a_{16}\) of the sequence, \(224,-112,56,-28,14,-7, \ldots\)
- Find \(a_{10}\) of the sequence, \(1,0.1,0.01,0.001,0.0001,0.00001, \ldots\)
- Find \(a_{9}\) of the sequence, \(1000,100,10,1,0.1,0.01, \dots\)
- Answer
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1. \(a_{9}=2,304 .\) The general term is \(a_{n}=9(2)^{n-1}\).
3. \(a_{15}=-\frac{2}{19,683} .\) The general term is \(a_{n}=-486\left(-\frac{1}{3}\right)^{n-1}\).
5. \(a_{10}=0.000000001 .\) The general term is \(a_{n}=(0.1)^{n-1}\).
Exercise \(\PageIndex{26}\) 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.
- \(8,24,72,216,648,1944, \dots\)
- \(7,14,28,56,112,224, \dots\)
- \(-6,12,-24,48,-96,192, \dots\)
- \(-4,12,-36,108,-324,972, \ldots\)
- \(81,27,9,3,1, \frac{1}{3}, \ldots\)
- \(256,64,16,4,1, \frac{1}{4}, \frac{1}{16}, \dots\)
- Answer
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1. \(57,395,624\)
3. \(-65,538\)
5. \(\frac{7,174,453}{59,049} \approx 121.5\)
Exercise \(\PageIndex{27}\) Find the Sum of the First \(n\) terms of a Geometric Sequence
In the following exercises, find the sum of the geometric sequence.
- \(\sum_{i=1}^{15}(2)^{i}\)
- \(\sum_{i=1}^{10}(3)^{i}\)
- \(\sum_{i=1}^{9} 4(2)^{i}\)
- \(\sum_{i=1}^{8} 5(3)^{i}\)
- \(\sum_{i=1}^{10} 9\left(\frac{1}{3}\right)^{i}\)
- \(\sum_{i=1}^{15} 4\left(\frac{1}{2}\right)^{i}\)
- Answer
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1. \(65,534\)
3. \(4088\)
5. \(\frac{29,524}{6561} \approx 4.5\)
Exercise \(\PageIndex{28}\) Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
- \(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}+\frac{1}{729}+\ldots\)
- \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\ldots\)
- \(6-2+\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\ldots\)
- \(-4+2-1+\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\dots\)
- \(6+12+24+48+96+192+\dots\)
- \(5+15+45+135+405+1215+\ldots\)
- \(1,024+512+256+128+64+32+\ldots\)
- \(6,561+2187+729+243+81+27+\dots\)
- Answer
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1. \(\frac{3}{2}\)
3. \(\frac{9}{2}\)
5. no sum as \(r≥1\)
7. \(2,048\(
Exercise \(\PageIndex{29}\) Find the Sum of an Infinite Geometric Series
In the following exercises, write each repeating decimal as a fraction.
- \(0 . \overline{3}\)
- \(0 . \overline{6}\)
- \(0 . \overline{7}\)
- \(0 . \overline{2}\)
- \(0 . \overline{45}\)
- \(0 . \overline{27}\)
- Answer
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1. \(\frac{1}{3}\)
3. \(\frac{7}{9}\)
5. \(\frac{5}{11}\)
Exercise \(\PageIndex{30}\) Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.
- 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. $\(1,500\) | \(80\)% |
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
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1. a. $\(6666.67\) b. $\(4000\) c. $\(15,000\) d. $\(7500\)
3. $\(295,581.88\)
5. $\(14,234.10\)
Exercise \(\PageIndex{31}\) Writing Exercises
- In your own words, explain how to determine whether a sequence is geometric.
- In your own words, explain how to find the general term of a geometric sequence.
- In your own words, explain the difference between a geometric sequence and a geometric series.
- In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.
- Answer
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2. Answers will vary.
4. Answers will vary
Self Check
- After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
- What does this checklist tell you about your mastery of this section? What steps will you take to improve?