24.3: Exercises

Exercise \(\PageIndex{1}\)

Which of these sequences is geometric, arithmetic, or neither or both. Write the sequence in the usual form \(a_n=a_1+d(n-1)\) if it is an arithmetic sequence and \(a_n=a_1\cdot r^{n-1}\) if it is a geometric sequence.

1. \(7, 14, 28, 56, \dots\)
2. \(3, -30, 300, -3000, 30000, \dots\)
3. \(81, 27, 9, 3, 1, \dfrac{1}{3}, \dots\)
4. \(-7, -5, -3, -1, 1, 3, 5, 7, \dots\)
5. \(-6, 2, -\dfrac 2 3, \dfrac 2 9, -\dfrac 2 {27}, \dots\)
6. \(-2, -2\cdot \dfrac 2 3, -2 \cdot \left(\dfrac 2 3\right)^2, -2 \cdot \left(\dfrac 2 3\right)^3, \dots\)
7. \(\dfrac 1 2, \dfrac 1 4, \dfrac 1 8, \dfrac 1 {16}, \dots\)
8. \(2, 2, 2, 2, 2, \dots\)
9. \(5, 1, 5, 1, 5, 1, 5, 1, \dots\)
10. \(-2, 2, -2, 2, -2, 2, \dots\)
11. \(0, 5, 10, 15, 20, \dots\)
12. \(5, \dfrac 5 3, \dfrac 5 {3^2}, \dfrac 5 {3^3}, \dfrac 5 {3^4}, \dots\)
13. \(\dfrac 1 2, \dfrac 1 4, \dfrac 1 8, \dfrac 1 {16}, \dots\)
14. \(\log(2), \log(4), \log(8), \log(16), \dots\)
15. \(a_n=-4^n\)
16. \(a_n=-4n\)
17. \(a_n=2\cdot (-9)^n\)
18. \(a_n=\left(\dfrac 1 3 \right)^n\)
19. \(a_n=-\left(\dfrac 5 7 \right)^n\)
20. \(a_n=\left(-\dfrac 5 7 \right)^n\)
21. \(a_n=\dfrac 2 n\)
22. \(a_n=3n+1\)

Answer

1. geometric, \(7 \cdot 2^{n-1}\)
2. geometric, \(3 \cdot(-10)^{n-1}\)
3. geometric, \(81\left(\dfrac{1}{3}\right)^{n-1}\)
4. arithmetic, \(-7+2(n-1)=-9+2 n\)
5. geometric, \(-6\left(-\dfrac{1}{3}\right)^{n-1}\)
6. geometric, \(-2\left(\dfrac{2}{3}\right)^{n-1}\)
7. geometric, \(\dfrac{1}{2}\left(\dfrac{1}{2}\right)^{n-1}\)
8. both, \(2=2+0(n-1)=2(1)^{n-1}\)
9. neither
10. geometric, \(-2(-1)^{n-1}\)
11. arithmetic, \(5(n − 1)\)
12. geometric, \(5\left(\dfrac{1}{3}\right)^{n-1}\)
13. geometric, \(\dfrac{1}{2}\left(\dfrac{1}{2}\right)^{n-1}\)
14. neither
15. geometric, \(-4(4)^{n-1}\)
16. arithmetic, \(−4 − 4(n − 1) = −4n\)
17. geometric, \(-18(-9)^{n-1}\)
18. geometric, \(\dfrac{1}{3}\left(\dfrac{1}{3}\right)^{n-1}\)
19. geometric, \(-\dfrac{5}{7}\left(\dfrac{5}{7}\right)^{n-1}\)
20. geometric, \(-\dfrac{5}{7}\left(-\dfrac{5}{7}\right)^{n-1}\)
21. neither
22. arithmetic, \(4+3(n-1)=1+3 n\)

Exercise \(\PageIndex{2}\)

A geometric sequence, \(a_n=a_1\cdot r^{n-1}\), has the given properties. Find the term \(a_n\) of the sequence.

1. \(a_1=3\), and \(r=5\), find \(a_4\)
2. \(a_1=200\), and \(r=-\dfrac 1 2\), find \(a_6\)
3. \(a_1=-7\), and \(r=2\), find \(a_n\) (for all \(n\))
4. \(r=2\), and \(a_4=48\), find \(a_1\)
5. \(r=100\), and \(a_{4}=900,000\), find \(a_n\) (for all \(n\))
6. \(a_{1}=20\), \(a_{4}=2500\), find \(a_n\) (for all \(n\))
7. \(a_1=\dfrac 1 8\), and \(a_6=\dfrac{3^5}{8^6}\), find \(a_n\) (for all \(n\))
8. \(a_3=36\), and \(a_{6}=972\), find \(a_n\) (for all \(n\))
9. \(a_{8}=4000\), \(a_{10}=40\), and \(r\) is negative, find \(a_n\) (for all \(n\))

Answer

1. \(375\)
2. \(6.25\)
3. \(-7 \cdot 2^{n-1}\)
4. \(6\)
5. \(\dfrac{9}{10}(100)^{n-1}\)
6. \(20 \cdot(5)^{n-1}\)
7. \(\dfrac{1}{8}\left(\dfrac{3}{8}\right)^{n-1}\)
8. \(4 \cdot 3^{n-1}\)
9. \(-40000000000\left(-\dfrac{1}{10}\right)^{n-1}\)

Exercise \(\PageIndex{3}\)

Find the value of the finite geometric series using formula [EQU:geometric-series]. Confirm the formula either by adding the the summands directly, or alternatively by using the calculator.

1. Find the sum \(\sum\limits_{j=1}^4 a_j\) for the geometric sequence \(a_j=5\cdot 4^{j-1}\).
2. Find the sum \(\sum\limits_{i=1}^7 a_i\) for the geometric sequence \(a_n=\left(\dfrac 1 2\right)^n\).
3. Find: \(\sum\limits_{m=1}^{5} \left(-\dfrac{1}{5}\right)^m\)
4. Find: \(\sum\limits_{k=1}^{6} 2.7\cdot 10^k\)
5. Find the sum of the first \(5\) terms of the geometric sequence: \[2, 6, 18, 54, \dots \nonumber \]
6. Find the sum of the first \(6\) terms of the geometric sequence: \[-5, 15, -45, 135, \dots \nonumber \]
7. Find the sum of the first \(8\) terms of the geometric sequence: \[-1, -7, -49, -343, \dots \nonumber \]
8. Find the sum of the first \(10\) terms of the geometric sequence: \[600, -300, 150, -75, 37.5, \dots \nonumber \]
9. Find the sum of the first \(40\) terms of the geometric sequence: \[5, 5, 5, 5, 5, \dots \nonumber \]

Answer

1. \(425\)
2. \(\dfrac{127}{128}\)
3. \(-\dfrac{521}{3125}\)
4. \(2999997\)
5. \(242\)
6. \(910\)
7. \(-960,800\)
8. \(\dfrac{25,575}{64}\)
9. \(200\)

Exercise \(\PageIndex{4}\)

Find the value of the infinite geometric series.

1. \(\sum_{j=1}^\infty a_j\), for \(a_j=3\cdot \left(\dfrac 2 3\right)^{j-1}\)
2. \(\sum_{j=1}^\infty 7\cdot \left(-\dfrac 1 5\right)^{j}\)
3. \(\sum_{j=1}^\infty 6\cdot \dfrac 1 {3^j}\)
4. \(\sum_{n=1}^\infty -2\cdot \left(0.8\right)^n\)
5. \(\sum_{n=1}^\infty \left(0.99\right)^n\)
6. \(27+9+3+1+\dfrac 1 3+\dots\)
7. \(-2+1-\dfrac 1 2+\dfrac 1 4-\dots\)
8. \(-6-2-\dfrac 2 3-\dfrac 2 9-\dots\)
9. \(100+40+16+6.4+ \dots\)
10. \(-54+18-6+2- \dots\)

Answer

1. \(9\)
2. \(-\dfrac{7}{6}\)
3. \(3\)
4. \(-8\)
5. \(99\)
6. \(\dfrac{81}{2}\)
7. \(-\dfrac{4}{3}\)
8. \(-9\)
9. \(\dfrac{500}{3}\)
10. \(-\dfrac{81}{2}\)

Exercise \(\PageIndex{5}\)

Rewrite the decimal using an infinite geometric sequence, and then use the formula for infinite geometric series to rewrite the decimal as a fraction (see example 24.2.3).

1. \(0.44444\dots\)
2. \(0.77777\dots\)
3. \(5.55555\dots\)
4. \(0.2323232323\dots\)
5. \(39.393939\dots\)
6. \(0.248248248\dots\)
7. \(20.02002\dots\)
8. \(0.5040504\dots\)

Answer

1. \(\dfrac{4}{9}\)
2. \(\dfrac{7}{9}\)
3. \(\dfrac{50}{9}\)
4. \(\dfrac{23}{99}\)
5. \(\dfrac{1300}{33}\)
6. \(\dfrac{248}{999}\)
7. \(\dfrac{20000}{999}\)
8. \(\dfrac{560}{1111}\)