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23.3: Exercises

Last updated

May 2, 2022
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- 23.2: The arithmetic sequence
- 24: The Geometric Series

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 $\PageIndex{1}$

Find the first seven terms of the sequence.

$a_n=3n$
$a_n=5n+3$
$a_n=n^2+2$
$a_n=n$
$a_n=(-1)^{n+1}$
$a_n=\dfrac{\sqrt{n+1}}{n}$
$a_k=10^k$
$a_i=5+(-1)^i$

Answer

$3, 6, 9, 12, 15, 18, 21$
$8, 13, 18, 23, 28, 33, 38$
$3, 6, 11, 18, 27, 38, 51$
$1, 2, 3, 4, 5, 6, 7$
$1, −1, 1, −1, 1, −1, 1$
$\sqrt{2}, \dfrac{\sqrt{3}}{2}, \dfrac{2}{3}, \dfrac{\sqrt{5}}{4}, \dfrac{\sqrt{6}}{5}, \dfrac{\sqrt{7}}{6}, \dfrac{\sqrt{8}}{7}$
$10, 100, 1000, 10000, 100000, 1000000, 10000000$
$4, 6, 4, 6, 4, 6, 4$

Exercise $\PageIndex{2}$

Find the first six terms of the sequence.

$a_1=5$ , $a_n=a_{n-1}+3$ for $n\geq 2$
$a_1=7$ , $a_n=10\cdot a_{n-1}$ for $n\geq 2$
$a_1=1$ , $a_n=2\cdot a_{n-1}+1$ for $n\geq 2$
$a_1=6$ , $a_2=4$ , $a_n=a_{n-1}-a_{n-2}$ for $n\geq 3$

Answer

$5, 8, 11, 14, 17$
$7, 70, 700, 7000, 70000$
$1, 3, 7, 15, 31$
$6, 4, −2, −6, −4$

Exercise $\PageIndex{3}$

Find the value of the series.

$\sum_{n=1}^4 a_n$ , where $a_n=5n$
$\sum_{k=1}^5 a_k$ , where $a_k=k$
$\sum_{i=1}^4 a_i$ , where $a_n=n^2$
$\sum_{n=1}^6 (n-4)$
$\sum_{k=1}^3 (k^2+4k-4)$
$\sum_{j=1}^4 \dfrac{1}{j+1}$

Answer

$50$
$15$
$30$
$−3$
$26$
$\dfrac {77}{60}$

Exercise $\PageIndex{4}$

Is the sequence below part of an arithmetic sequence? In the case that it is part of an arithmetic sequence, find the formula for the $n$ th term $a_n$ in the form $a_n=a_1+d\cdot (n-1)$ .

$5, 8, 11, 14, 17, \dots$
$-10, -7, -4, -1, 2, \dots$
$-1, 1, -1, 1, -1, 1, \dots$
$18, 164, 310, 474, \dots$
$73.4, 51.7, 30, \dots$
$9, 3, -3, -8, -14, \dots$
$4, 4, 4, 4, 4, \dots$
$-2.72, -2.82, -2.92, -3.02, -3.12, \dots$
$\sqrt{2}, \sqrt{5}, \sqrt{8}, \sqrt{11}, \dots$
$\dfrac{-3}{5}, \dfrac{-1}{10}, \dfrac{2}{5}, \dots$
$a_n=4+5\cdot n$
$a_j=2\cdot j-5$
$a_n=n^2 +8n+15$
$a_k=9\cdot (k+5) +7k-1$

Answer

For the convenience of those who prefer to use $a_{n}=a+b \cdot n$ as standard form we have provided answers also in that form.

$5 + 3(n−1) = 2 + 3n$
$−10 + 3(n−1) = −13 + 3n$
no
no
$73.4−21.7(n−1) = 95.1−21.7n$
no
$4 + 0 ·(n−1) = 4 + 0 ·n$
$-2.72-.1(n-1)=-2.62-.1 n$
no
$-\dfrac{3}{5}+\dfrac{1}{2}(n-1)=-\dfrac{11}{10}+\dfrac{1}{2} n$
$9 + 5(n − 1) = 4 + 5n$
$−3 + 2(j − 1) = −5 + 2j$
no
$29 + 16(k − 1) = 13 + 16k$

Exercise $\PageIndex{5}$

Determine the general $n$ th term $a_n$ of an arithmetic sequence $\{a_n\}$ with the data given below.

$d=4$ , and $a_{8}=57$
$d=-3$ , and $a_{99}=-70$
$a_1=14$ , and $a_{7}=-16$
$a_1=-80$ , and $a_{5}=224$
$a_{3}=10$ , and $a_{14}=-23$
$a_{20}=2$ , and $a_{60}=32$

Answer

$57 + 4(n − 8) = 29 + 4(n − 1) = 25 + 4n$
$−70 − 3(n − 99) = 224 − 3(n − 1) = 227 − 3n$
$14 − 5(n − 1) = 19 − 5n$
$−80 + 76(n − 1) = −156 + 76n$
$10 − 3(n − 3) = 16 − 3(n − 1) = 19 − 3n$
$2+\dfrac{3}{4}(n-20)=-49 / 4+\dfrac{3}{4}(n-1)=-13+\dfrac{3}{4} n$

Exercise $\PageIndex{6}$

Determine the value of the indicated term of the given arithmetic sequence.

if $a_1=8$ , and $a_{15}=92$ , find $a_{19}$
if $d=-2$ , and $a_3=31$ , find $a_{81}$
if $a_1=0$ , and $a_{17}=-102$ , find $a_{73}$
if $a_{7}=128$ , and $a_{37}=38$ , find $a_{26}$

Answer

$116$
$187$
$-\dfrac{3621}{8}$
$71$

Exercise $\PageIndex{7}$

Determine the sum of the arithmetic sequence.

Find the sum $a_1+\dots +a_{48}$ for the arithmetic sequence $a_i=4i+7$ .
Find the sum $\sum_{i=1}^{21}a_i$ for the arithmetic sequence $a_n=2-5n$ .
Find the sum: $\sum\limits_{i=1}^{99} (10\cdot i+1)$
Find the sum: $\sum\limits_{n=1}^{200} (-9-n)$
Find the sum of the first terms of the arithmetic sequence: $2, 4, 6, 8, 10, 12, \dots$
Find the sum of the first terms of the arithmetic sequence: $25, 21, 17, 13, 9, 5, \dots$
Find the sum of the first terms of the arithmetic sequence: $2012, 2002, 1992, 1982, \dots$
Find the sum of the first terms of the arithmetic sequence: $-11, -6, -1, \dots$
Find the sum of the first terms of the arithmetic sequence: $-8, -8.2, -8.4, -8.6, -8.8, -9, -9.2, \dots$
Find the sum $7+8+9+10+\dots+776+777$
Find the sum of the first terms of the arithmetic sequence: $5, 5, 5, 5, 5, \dots$

Answer

$5, 040$
$−1, 113$
$49, 599$
$−21, 900$
$10, 100$
$−11, 537$
$123, 150$
$424$
$−1762.2$
$302, 232$
$200$

Exercise 23.3.1\PageIndex{1}

Exercise 23.3.2\PageIndex{2}

Exercise 23.3.3\PageIndex{3}

Exercise 23.3.4\PageIndex{4}

Exercise 23.3.5\PageIndex{5}

Exercise 23.3.6\PageIndex{6}

Exercise 23.3.7\PageIndex{7}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$

Exercise $\PageIndex{6}$

Exercise $\PageIndex{7}$