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24.3: Exercises

Last updated

May 2, 2022
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- 24.2: Infinite Geometric Series
- 25: The Binomial Theorem

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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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.

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

Answer

geometric, $7 \cdot 2^{n-1}$
geometric, $3 \cdot(-10)^{n-1}$
geometric, $81\left(\dfrac{1}{3}\right)^{n-1}$
arithmetic, $-7+2(n-1)=-9+2 n$
geometric, $-6\left(-\dfrac{1}{3}\right)^{n-1}$
geometric, $-2\left(\dfrac{2}{3}\right)^{n-1}$
geometric, $\dfrac{1}{2}\left(\dfrac{1}{2}\right)^{n-1}$
both, $2=2+0(n-1)=2(1)^{n-1}$
neither
geometric, $-2(-1)^{n-1}$
arithmetic, $5(n − 1)$
geometric, $5\left(\dfrac{1}{3}\right)^{n-1}$
geometric, $\dfrac{1}{2}\left(\dfrac{1}{2}\right)^{n-1}$
neither
geometric, $-4(4)^{n-1}$
arithmetic, $−4 − 4(n − 1) = −4n$
geometric, $-18(-9)^{n-1}$
geometric, $\dfrac{1}{3}\left(\dfrac{1}{3}\right)^{n-1}$
geometric, $-\dfrac{5}{7}\left(\dfrac{5}{7}\right)^{n-1}$
geometric, $-\dfrac{5}{7}\left(-\dfrac{5}{7}\right)^{n-1}$
neither
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.

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

Answer

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

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.

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

Answer

$425$
$\dfrac{127}{128}$
$-\dfrac{521}{3125}$
$2999997$
$242$
$910$
$-960,800$
$\dfrac{25,575}{64}$
$200$

Exercise $\PageIndex{4}$

Find the value of the infinite geometric series.

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

Answer

$9$
$-\dfrac{7}{6}$
$3$
$-8$
$99$
$\dfrac{81}{2}$
$-\dfrac{4}{3}$
$-9$
$\dfrac{500}{3}$
$-\dfrac{81}{2}$

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).

$0.44444\dots$
$0.77777\dots$
$5.55555\dots$
$0.2323232323\dots$
$39.393939\dots$
$0.248248248\dots$
$20.02002\dots$
$0.5040504\dots$

Answer

$\dfrac{4}{9}$
$\dfrac{7}{9}$
$\dfrac{50}{9}$
$\dfrac{23}{99}$
$\dfrac{1300}{33}$
$\dfrac{248}{999}$
$\dfrac{20000}{999}$
$\dfrac{560}{1111}$

Exercise 24.3.1\PageIndex{1}

Exercise 24.3.2\PageIndex{2}

Exercise 24.3.3\PageIndex{3}

Exercise 24.3.4\PageIndex{4}

Exercise 24.3.5\PageIndex{5}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$