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

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. \(a_{n}=(-1)^{n+1}(n-1)\)

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.

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 \(a_{2}=0\)

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

Answer

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. \(\sum_{k=-2}^{2}\left(\frac{2}{3}\right)^{k}\)

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.

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. \(a_{1}=1 ; d=1\)

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.

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 \(a_{7}=1\)

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.

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. \(a_{n}=-6.5 n+3 ; S_{35}\)

Answer

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?

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.

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?

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.

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. \(-1, \frac{2}{3},-\frac{4}{9}, \ldots\)

Answer

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. \(\sum_{n=1}^{31} 3(-1)^{n-1}\)

Answer

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. \(\frac{(n-2) !}{(n+1) !}\)

Answer

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. \(\left( \begin{array}{c}{n} \\ {n-2}\end{array}\right)\)

Answer

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. \(\left(x^{3}-2 y^{2}\right)^{4}\)

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.

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

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

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

Answer

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. \(\frac{1}{2}, \frac{1}{4}, \frac{2}{9}, \ldots\)

Answer

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 \(a_{9}=2\)

Answer

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 \(a_{8}=-32\)

Answer

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. \(\sum_{n=1}^{100}\left(2 n-\frac{3}{2}\right)\)

Answer

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.

Answer

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. \(\left(3 a+b^{2}\right)^{5}\)

Answer

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