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

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Exercise $$\PageIndex{1}$$

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

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

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.

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

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

1. $$35$$

3. $$-5$$

5. $$-18$$

Exercise $$\PageIndex{4}$$

Evaluate.

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

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.

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

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.

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

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.

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

1. $$7,140$$

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

5. $$1,674$$

Exercise $$\PageIndex{8}$$

Evaluate.

1. $$\sum_{n=1}^{22}(7 n-5)$$
2. $$\sum_{n=1}^{100}(1-4 n)$$
3. $$\sum_{n=1}^{35}\left(\frac{2}{3} n\right)$$
4. $$\sum_{n=1}^{30}\left(-\frac{1}{4} n+1\right)$$
5. $$\sum_{n=1}^{40}(2.3 n-1.1)$$
6. $$\sum_{n=1}^{300} n$$
7. Find the sum of the first $$175$$ positive odd integers.
8. Find the sum of the first $$175$$ positive even integers.
9. Find all arithmetic means between $$a_{1} = \frac{2}{3}$$ and $$a_{5} = −\frac{2}{3}$$
10. Find all arithmetic means between $$a_{3} = −7$$ and $$a_{7} = 13$$.
11. 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.
12. 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?

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. 1. $$a_{1}=5 ; r=2$$ 2. $$a_{1}=3 ; r=-2$$ 3. $$a_{1}=1 ; r=-\frac{3}{2}$$ 4. $$a_{1}=-4 ; r=\frac{1}{3}$$ 5. $$a_{1}=1.2 ; r=0.2$$ 6. $$a_{1}=-5.4 ; r=-0.1$$ 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. 1. $$4, 40, 400,…$$ 2. $$−6, −30, −150,…$$ 3. $$6, \frac{9}{2}, \frac{27}{8}, \dots$$ 4. $$1, \frac{3}{5}, \frac{9}{25}, \dots$$ 5. $$a_{4}=-4$$ and $$a_{9}=128$$ 6. $$a_{2}=-1$$ and $$a_{5}=-64$$ 7. $$a_{2}=-\frac{5}{2}$$ and $$a_{5}=-\frac{625}{16}$$ 8. $$a_{3}=50$$ and $$a_{6}=-6,250$$ 9. Find all geometric means between $$a_{1} = −1$$ and $$a_{4} = 64$$. 10. 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. 1. $$a_{n}=3(4)^{n-1} ; S_{6}$$ 2. $$a_{n}=-5(3)^{n-1} ; S_{10}$$ 3. $$a_{n}=\frac{3}{2}(-2)^{n} ; S_{14}$$ 4. $$a_{n}=\frac{1}{5}(-3)^{n+1} ; S_{12}$$ 5. $$a_{n}=8\left(\frac{1}{2}\right)^{n+2} ; S_{8}$$ 6. $$a_{n}=\frac{1}{8}(-2)^{n+2} ; S_{10}$$ Answer 1. $$4,095$$ 3. $$16,383$$ 5. $$\frac{255}{128}$$ Exercise $$\PageIndex{12}$$ Evaluate. 1. $$\sum_{n=1}^{10} 3(-4)^{n}$$ 2. $$\sum_{n=1}^{9}-\frac{3}{5}(-2)^{n-1}$$ 3. $$\sum_{n=1}^{\infty}-3\left(\frac{2}{3}\right)^{n}$$ 4. $$\sum_{n=1}^{\infty} \frac{1}{2}\left(\frac{4}{5}\right)^{n+1}$$ 5. $$\sum_{n=1}^{\infty} \frac{1}{2}\left(-\frac{3}{2}\right)^{n}$$ 6. $$\sum_{n=1}^{\infty} \frac{3}{2}\left(-\frac{1}{2}\right)^{n}$$ 7. 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.
8. 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.
9. 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.
10. 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?

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.

1. $$4, 9, 14,…$$
2. $$6, 18, 54,…$$
3. $$-1,-\frac{1}{2}, 0, \dots$$
4. $$10,30,60, \dots$$
5. $$0,1,8, \dots$$
6. $$-1, \frac{2}{3},-\frac{4}{9}, \ldots$$

1. Arithmetic; $$d=5$$

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

5. Neither

Exercise $$\PageIndex{14}$$

Evaluate.

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

1. $$30$$

3. $$−1,952$$

5. $$1,640$$

7. $$−187$$

9. $$84$$

11. $$3$$

Exercise $$\PageIndex{15}$$

Evaluate.

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

2. $$39,916,800$$

4. $$54$$

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

Exercise $$\PageIndex{16}$$

Calculate the indicated binomial coefficient.

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

2. $$56$$

4. $$11$$

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

Exercise $$\PageIndex{17}$$

Expand using the binomial theorem.

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

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.

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

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

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

1. $$70$$

Exercise $$\PageIndex{20}$$

Classify the sequence as arithmetic, geometric, or neither.

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

1. Arithmetic

3. Geometric

Exercise $$\PageIndex{21}$$

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

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

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

Exercise $$\PageIndex{22}$$

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

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

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

Exercise $$\PageIndex{23}$$

Calculate the indicated sum.

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

1. $$-770$$

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

Exercise $$\PageIndex{24}$$

Evaluate.

1. $$\frac{14 !}{10 ! 6 !}$$
2. $$\left( \begin{array}{l}{9} \\ {7}\end{array}\right)$$
3. Determine the sum of the first $$48$$ positive odd integers.
4. 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?
5. 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.

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

3. $$2,304$$

5. $$54$$ feet

Exercise $$\PageIndex{25}$$

Expand using the binomial theorem.

1. $$(x-5 y)^{4}$$
2. $$\left(3 a+b^{2}\right)^{5}$$
2. $$\begin{array}{l}{243 a^{5}+405 a^{4} b^{2}+270 a^{3} b^{4}} {+90 a^{2} b^{6}+15 a b^{8}+b^{10}}\end{array}$$