9.E: Sequences, Series, and the Binomial Theorem (Exercises)

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- 9.4: Binomial Theorem
- Back Matter

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Exercise $\PageIndex{1}$

Find the first $5$ terms of the sequence as well as the $30^{th}$ term.

$a_{n}=5 n-3$
$a_{n}=-4 n+3$
$a_{n}=-10 n$
$a_{n}=3 n$
$a_{n}=(-1)^{n}(n-2)^{2}$
$a_{n}=\frac{(-1)^{n}}{2 n-1}$
$a_{n}=\frac{2 n+1}{n}$
$a_{n}=(-1)^{n+1}(n-1)$

Answer

1. $2,7,12,17,22 ; a_{30}=147$

3. $-10,-20,-30,-40,-50 ; a_{30}=-300$

5. $-1,0,-1,4,-9 ; a_{30}=784$

7. $3, \frac{5}{2}, \frac{7}{3}, \frac{9}{4}, \frac{11}{5} ; a_{30}=\frac{61}{30}$

Exercise $\PageIndex{2}$

Find the first $5$ terms of the sequence.

$a_{n}=\frac{n x^{n}}{2 n+1}$
$a_{n}=\frac{(-1)^{n-1} x^{n+2}}{n}$
$a_{n}=2^{n} x^{2 n}$
$a_{n}=(-3 x)^{n-1}$
$a_{n}=a_{n-1}+5$ where $a_{1}=0$
$a_{n}=4 a_{n-1}+1$ where $a_{1}=-2$
$a_{n}=a_{n-2}-3 a_{n-1}$ where $a_{1}=0$ and $a_{2}=-3$
$a_{n}=5 a_{n-2}-a_{n-1}$ where $a_{1}=-1$ and $a_{2}=0$

Answer

1. $\frac{x}{3}, \frac{2 x^{2}}{5}, \frac{3 x^{3}}{7}, \frac{4 x^{4}}{9}, \frac{5 x^{5}}{11}$

3. $2 x^{2}, 4 x^{4}, 8 x^{6}, 16 x^{8}, 32 x^{10}$

5. $0, 5, 10, 15, 20$

7. $0, −3, 9, −30, 99$

Exercise $\PageIndex{3}$

Find the indicated partial sum.

$1,4,7,10,13, \dots ; S_{5}$
$3,1,-1,-3,-5, \dots ; S_{5}$
$-1,3,-5,7,-9, \ldots ; S_{4}$
$a_{n}=(-1)^{n} n^{2} ; S_{4}$
$a_{n}=-3(n-2)^{2} ; S_{4}$
$a_{n}=\left(-\frac{1}{5}\right)^{n-2} ; S_{4}$

Answer

1. $35$

3. $-5$

5. $-18$

Exercise $\PageIndex{4}$

Evaluate.

$\sum_{k=1}^{6}(1-2 k)$
$\sum_{k=1}^{4}(-1)^{k} 3 k^{2}$
$\sum_{n=1}^{3} \frac{n+1}{n}$
$\sum_{n=1}^{7} 5(-1)^{n-1}$
$\sum_{k=4}^{8}(1-k)^{2}$
$\sum_{k=-2}^{2}\left(\frac{2}{3}\right)^{k}$

Answer

1. $-36$

3. $\frac{29}{6}$

5. $135$

Exercise $\PageIndex{5}$

Write the first $5$ terms of the arithmetic sequence given its first term and common difference. Find a formula for its general term.

$a_{1}=6 ; d=5$
$a_{1}=5 ; d=7$
$a_{1}=5 ; d=-3$
$a_{1}=-\frac{3}{2} ; d=-\frac{1}{2}$
$a_{1}=-\frac{3}{4} ; d=-\frac{3}{4}$
$a_{1}=-3.6 ; d=1.2$
$a_{1}=7 ; d=0$
$a_{1}=1 ; d=1$

Answer

1. $6,11,16,21,26 ; a_{n}=5 n+1$

3. $5,2,-1,-4,-7 ; a_{n}=8-3 n$

5. $-\frac{3}{4},-\frac{3}{2},-\frac{9}{4},-3,-\frac{15}{4} ; a_{n}=-\frac{3}{4} n$

7. $7,7,7,7,7 ; a_{n}=7$

Exercise $\PageIndex{6}$

Given the terms of an arithmetic sequence, find a formula for the general term.

$10, 20, 30, 40, 50,…$
$−7, −5, −3, −1, 1,…$
$−2, −5, −8, −11, −14,…$
$-\frac{1}{3}, 0, \frac{1}{3}, \frac{2}{3}, 1, \ldots$
$a_{4}=11$ and $a_{9}=26$
$a_{5}=-5$ and $a_{10}=-15$
$a_{6}=6$ and $a_{24}=15$
$a_{3}=-1.4$ and $a_{7}=1$

Answer

1. $a_{n}=10 n$

3. $a_{n}=1-3 n$

5. $a_{n}=3 n-1$

7. $a_{n}=\frac{1}{2} n+3$

Exercise $\PageIndex{7}$

Calculate the indicated sum given the formula for the general term of an arithmetic sequence.

$a_{n}=4 n-3 ; S_{60}$
$a_{n}=-2 n+9 ; S_{35}$
$a_{n}=\frac{1}{5} n-\frac{1}{2}; S_{15}$
$a_{n}=-n+\frac{1}{4} ; S_{20}$
$a_{n}=1.8 n-4.2 ; S_{45}$
$a_{n}=-6.5 n+3 ; S_{35}$

Answer

1. $7,140$

3. $\frac{33}{2}$

5. $1,674$

Exercise $\PageIndex{8}$

Evaluate.

$\sum_{n=1}^{22}(7 n-5)$
$\sum_{n=1}^{100}(1-4 n)$
$\sum_{n=1}^{35}\left(\frac{2}{3} n\right)$
$\sum_{n=1}^{30}\left(-\frac{1}{4} n+1\right)$
$\sum_{n=1}^{40}(2.3 n-1.1)$
$\sum_{n=1}^{300} n$
Find the sum of the first $175$ positive odd integers.
Find the sum of the first $175$ positive even integers.
Find all arithmetic means between $a_{1} = \frac{2}{3}$ and $a_{5} = −\frac{2}{3}$
Find all arithmetic means between $a_{3} = −7$ and $a_{7} = 13$ .
A $5$ -year salary contract offers $ $58,200$ for the first year with a $ $4,200$ increase each additional year. Determine the total salary obligation over the $5$ -year period.
The first row of seating in a theater consists of $10$ seats. Each successive row consists of four more seats than the previous row. If there are $14$ rows, how many total seats are there in the theater?

Answer

1. $1,661$

3. $420$

5. $1,842$

7. $30,625$

9. $\frac{1}{3}, 0, −\frac{1}{3}$

11. $ $333,000$

Exercise $\PageIndex{9}$

Write the first $5$ terms of the geometric sequence given its first term and common ratio. Find a formula for its general term.

$a_{1}=5 ; r=2$
$a_{1}=3 ; r=-2$
$a_{1}=1 ; r=-\frac{3}{2}$
$a_{1}=-4 ; r=\frac{1}{3}$
$a_{1}=1.2 ; r=0.2$
$a_{1}=-5.4 ; r=-0.1$

Answer

1. $5,10,20,40,80 ; a_{n}=5(2)^{n-1}$

3. $1,-\frac{3}{2}, \frac{9}{4},-\frac{27}{8}, \frac{81}{16} ; a_{n}=\left(-\frac{3}{2}\right)^{n-1}$

5. $1.2,0.24,0.048,0.0096,0.00192 ; a_{n}=1.2(0.2)^{n-1}$

Exercise $\PageIndex{10}$

Given the terms of a geometric sequence, find a formula for the general term.

$4, 40, 400,…$
$−6, −30, −150,…$
$6, \frac{9}{2}, \frac{27}{8}, \dots$
$1, \frac{3}{5}, \frac{9}{25}, \dots$
$a_{4}=-4$ and $a_{9}=128$
$a_{2}=-1$ and $a_{5}=-64$
$a_{2}=-\frac{5}{2}$ and $a_{5}=-\frac{625}{16}$
$a_{3}=50$ and $a_{6}=-6,250$
Find all geometric means between $a_{1} = −1$ and $a_{4} = 64$ .
Find all geometric means between $a_{3} = 6$ and $a_{6} = 162$ .

Answer

1. $a_{n}=4(10)^{n-1}$

3. $a_{n}=6\left(\frac{3}{4}\right)^{n-1}$

5. $a_{1}=\frac{1}{2}(-2)^{n-1}$

7. $a_{n}=-\left(\frac{5}{2}\right)^{n-1}$

9. $4, 16$

Exercise $\PageIndex{11}$

Calculate the indicated sum given the formula for the general term of a geometric sequence.

$a_{n}=3(4)^{n-1} ; S_{6}$
$a_{n}=-5(3)^{n-1} ; S_{10}$
$a_{n}=\frac{3}{2}(-2)^{n} ; S_{14}$
$a_{n}=\frac{1}{5}(-3)^{n+1} ; S_{12}$
$a_{n}=8\left(\frac{1}{2}\right)^{n+2} ; S_{8}$
$a_{n}=\frac{1}{8}(-2)^{n+2} ; S_{10}$

Answer

1. $4,095$

3. $16,383$

5. $\frac{255}{128}$

Exercise $\PageIndex{12}$

Evaluate.

$\sum_{n=1}^{10} 3(-4)^{n}$
$\sum_{n=1}^{9}-\frac{3}{5}(-2)^{n-1}$
$\sum_{n=1}^{\infty}-3\left(\frac{2}{3}\right)^{n}$
$\sum_{n=1}^{\infty} \frac{1}{2}\left(\frac{4}{5}\right)^{n+1}$
$\sum_{n=1}^{\infty} \frac{1}{2}\left(-\frac{3}{2}\right)^{n}$
$\sum_{n=1}^{\infty} \frac{3}{2}\left(-\frac{1}{2}\right)^{n}$
After the first year of operation, the value of a company van was reported to be $ $40,000$ . Because of depreciation, after the second year of operation the van was reported to have a value of $ $32,000$ and then $ $25,600$ after the third year of operation. Write a formula that gives the value of the van after the $n$ th year of operation. Use it to determine the value of the van after $10$ years of operation.
The number of cells in a culture of bacteria doubles every $6$ hours. If $250$ cells are initially present, write a sequence that shows the number of cells present after every $6$ -hour period for one day. Write a formula that gives the number of cells after the $n$ th $6$ -hour period.
A ball bounces back to one-half of the height that it fell from. If dropped from $32$ feet, approximate the total distance the ball travels.
A structured settlement yields an amount in dollars each year $n$ according to the formula $p_{n}=12,500(0.75)^{n-1}$ . What is the total value of a $10$ -year settlement?

Answer

1. $2,516,580$

3. $−6$

5. No sum

7. $v_{n}=40,000(0.8)^{n-1} ; v_{10}=\$ 5,368.71$

9. $96$ feet

Exercise $\PageIndex{13}$

Classify the sequence as arithmetic, geometric, or neither.

$4, 9, 14,…$
$6, 18, 54,…$
$-1,-\frac{1}{2}, 0, \dots$
$10,30,60, \dots$
$0,1,8, \dots$
$-1, \frac{2}{3},-\frac{4}{9}, \ldots$

Answer

1. Arithmetic; $d=5$

3. Arithmetic; $d=\frac{1}{2}$

5. Neither

Exercise $\PageIndex{14}$

Evaluate.

$\sum_{n=1}^{4} n^{2}$
$\sum_{n=1}^{4} n^{3}$
$\sum_{n=1}^{32}(-4 n+5)$
$\sum_{n=1}^{\infty}-2\left(\frac{1}{5}\right)^{n-1}$
$\sum_{n=1}^{8} \frac{1}{3}(-3)^{n}$
$\sum_{n=1}^{46}\left(\frac{1}{4} n-\frac{1}{2}\right)$
$\sum_{n=1}^{22}(3-n)$
$\sum_{n=1}^{31} 2 n$
$\sum_{n=1}^{28} 3$
$\sum_{n=1}^{30} 3(-1)^{n-1}$
$\sum_{n=1}^{31} 3(-1)^{n-1}$

Answer

1. $30$

3. $−1,952$

5. $1,640$

7. $−187$

9. $84$

11. $3$

Exercise $\PageIndex{15}$

Evaluate.

$8!$
$11!$
$\frac{10 !}{2 ! 6 !}$
$\frac{9 ! 3 !}{8 !}$
$\frac{(n+3) !}{n !}$
$\frac{(n-2) !}{(n+1) !}$

Answer

2. $39,916,800$

4. $54$

6. $\frac{1}{n(n+1)(n-1)}$

Exercise $\PageIndex{16}$

Calculate the indicated binomial coefficient.

$\left( \begin{array}{l}{7} \\ {4}\end{array}\right)$
$\left( \begin{array}{l}{8} \\ {3}\end{array}\right)$
$\left( \begin{array}{c}{10} \\ {5}\end{array}\right)$
$\left( \begin{array}{l}{11} \\ {10}\end{array}\right)$
$\left( \begin{array}{c}{12} \\ {0}\end{array}\right)$
$\left( \begin{array}{l}{n+1} \\ {n-1}\end{array}\right)$
$\left( \begin{array}{c}{n} \\ {n-2}\end{array}\right)$

Answer

2. $56$

4. $11$

6. $\frac{n(n+1)}{2}$

Exercise $\PageIndex{17}$

Expand using the binomial theorem.

$(x+7)^{3}$
$(x-9)^{3}$
$(2 y-3)^{4}$
$(y+4)^{4}$
$(x+2 y)^{5}$
$(3 x-y)^{5}$
$(u-v)^{6}$
$(u+v)^{6}$
$\left(5 x^{2}+2 y^{2}\right)^{4}$
$\left(x^{3}-2 y^{2}\right)^{4}$

Answer

1. $x^{3}+21 x^{2}+147 x+343$

3. $16 y^{4}-96 y^{3}+216 y^{2}-216 y+81$

5. $x^{5}+10 x^{4} y+40 x^{3} y^{2}+80 x^{2} y^{3}+80 x y^{4}+32 y^{5}$

7. $\begin{array}{l}{u^{6}-6 u^{5} v+15 u^{4} v^{2}-20 u^{3} v^{3}} {+15 u^{2} v^{4}-6 u v^{5}+v^{6}}\end{array}$

9. $625 x^{8}+1,000 x^{6} y^{2}+600 x^{4} y^{4}+160 x^{2} y^{6}+16 y^{8}$

Sample Exam

Exercise $\PageIndex{18}$

Find the first $5$ terms of the sequence.

$a_{n}=6 n-15$
$a_{n}=5(-4)^{n-2}$
$a_{n}=\frac{n-1}{2 n-1}$
$a_{n}=(-1)^{n-1} x^{2 n}$

Answer

1. $-9,-3,3,9,15$

3. $0, \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}$

Exercise $\PageIndex{19}$

Find the indicated partial sum

$a_{n}=(n-1) n^{2} ; S_{4}$
$\sum_{k=1}^{5}(-1)^{k} 2^{k-2}$

Answer: 1. $70$

Exercise $\PageIndex{20}$

Classify the sequence as arithmetic, geometric, or neither.

$-1,-\frac{3}{2},-2, \ldots$
$1,-6,36, \dots$
$\frac{3}{8},-\frac{3}{4}, \frac{3}{2}, \ldots$
$\frac{1}{2}, \frac{1}{4}, \frac{2}{9}, \ldots$

Answer

1. Arithmetic

3. Geometric

Exercise $\PageIndex{21}$

Given the terms of an arithmetic sequence, find a formula for the general term.

$10,5,0,-5,-10, \dots$
$a_{4}=-\frac{1}{2}$ and $a_{9}=2$

Answer: 1. $a_{n}=15-5 n$

Exercise $\PageIndex{22}$

Given the terms of a geometric sequence, find a formula for the general term.

$-\frac{1}{8},-\frac{1}{2},-2,-8,-32, \ldots$
$a_{3}=1$ and $a_{8}=-32$

Answer: 1. $a_{n}=-\frac{1}{8}(4)^{n-1}$

Exercise $\PageIndex{23}$

Calculate the indicated sum.

$a_{n}=5-n ; S_{44}$
$a_{n}=(-2)^{n+2} ; S_{12}$
$\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n-1}$
$\sum_{n=1}^{100}\left(2 n-\frac{3}{2}\right)$

Answer

1. $-770$

3. $\frac{8}{3}$

Exercise $\PageIndex{24}$

Evaluate.

$\frac{14 !}{10 ! 6 !}$
$\left( \begin{array}{l}{9} \\ {7}\end{array}\right)$
Determine the sum of the first $48$ positive odd integers.
The first row of seating in a theater consists of $14$ seats. Each successive row consists of two more seats than the previous row. If there are $22$ rows, how many total seats are there in the theater?
A ball bounces back to one-third of the height that it fell from. If dropped from $27$ feet, approximate the total distance the ball travels.

Answer

1. $\frac{1,001}{30}$

3. $2,304$

5. $54$ feet

Exercise $\PageIndex{25}$

Expand using the binomial theorem.

$(x-5 y)^{4}$
$\left(3 a+b^{2}\right)^{5}$

Answer: 2. $\begin{array}{l}{243 a^{5}+405 a^{4} b^{2}+270 a^{3} b^{4}} {+90 a^{2} b^{6}+15 a b^{8}+b^{10}}\end{array}$

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