# 24.3: Exercises

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## 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. $$a_n=3n+1$$
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 $$n$$)
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: $5, 5, 5, 5, 5, \dots \nonumber$
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. $$-54+18-6+2- \dots$$
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. $$0.5040504\dots$$
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. $$\dfrac{560}{1111}$$