Chapter 12 Review Exercises

Sequences

Exercise $\PageIndex{1}$ Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

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

Answer

2. $7,13,31,85,247$

4. $\frac{3}{4}, \frac{5}{16}, \frac{7}{64}, \frac{9}{256}, \frac{11}{1024}$

Exercise $\PageIndex{2}$ Find a Formula for the General Term ($n$th Term of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

1. $9,18,27,36,45, \dots$
2. $-5,-4,-3,-2,-1, \dots$
3. $\frac{1}{e^{3}}, \frac{1}{e^{2}}, \frac{1}{e}, 1, e, \ldots$
4. $1,-8,27,-64,125, \ldots$
5. $-\frac{1}{3},-\frac{1}{2},-\frac{3}{5},-\frac{2}{3},-\frac{5}{7}, \dots$

Answer

1. $a_{n}=9 n$

3. $a_{n}=e^{n-4}$

5. $a_{n}=-\frac{n}{n+2}$

Exercise $\PageIndex{3}$ Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

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

Answer

2. $\frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, \frac{1}{42}$

Exercise $\PageIndex{4}$ Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

1. $\sum_{i=1}^{7}(2 i-5)$
2. $\sum_{i=1}^{3} 5^{i}$
3. $\sum_{k=0}^{4} \frac{4}{k !}$
4. $\sum_{k=1}^{4}(k+1)(2 k+1)$

Answer

1. $\begin{array}{l}{-3+(-1)+1+3+5} {+7+9=21}\end{array}$

3. $4+4+2+\frac{2}{3}+\frac{1}{6}=\frac{65}{6}$

Exercise $\PageIndex{5}$ Use Summation Notation to Write a Sum

In the following exercises, write each sum using summation notation.

1. $-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}$
2. $4-8+12-16+20-24$
3. $4+2+\frac{4}{3}+1+\frac{4}{5}$

Answer

1. $\sum_{n=1}^{5}(-1)^{n} \frac{1}{3^{n}}$

3. $\sum_{n=1}^{5} \frac{4}{n}$

Arithmetic Sequences

Exercise $\PageIndex{6}$ Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

1. $1,2,4,8,16,32, \dots$
2. $-7,-1,5,11,17,23, \dots$
3. $13,9,5,1,-3,-7, \dots$

Answer

2. The sequence is arithmetic with common difference $d=6$.

Exercise $\PageIndex{7}$ Determine if a Sequence is Arithmetic

In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.

1. $a_{1}=5$ and $d=3$
2. $a_{1}=8$ and $d=-2$
3. $a_{1}=-13$ and $d=6$

Answer

1. $5,8,11,14,17$

3. $-13,-7,-1,5,11$

Exercise $\PageIndex{8}$ Find the General Term ($n$th Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

1. Find the twenty-fifth term of a sequence where the first term is five and the common difference is three.
2. Find the thirtieth term of a sequence where the first term is $16$ and the common difference is $−5$.
3. Find the seventeenth term of a sequence where the first term is $−21$ and the common difference is two.

Answer

2. $-129$

Exercise $\PageIndex{9}$ Find the General Term ($n$th Term) of an Arithmetic Sequence

In the following exercises, find the indicated term and give the formula for the general term.

1. Find the eighteenth term of a sequence where the fifth term is $12$ and the common difference is seven.
2. Find the twenty-first term of a sequence where the seventh term is $14$ and the common difference is $−3$.

Answer

1. $a_{18}=103 .$ The general term is $a_{n}=7 n-23$.

Exercise $\PageIndex{10}$ Find the General Term ($n$th Term) of an Arithmetic Sequence

In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

1. The fifth term is $17$ and the fourteenth term is $53$.
2. The third term is $−26$ and the sixteenth term is $−91$.

Answer

1. $a_{1}=1, d=4 .$ The general term is $a_{n}=4 n-3$.

Exercise $\PageIndex{11}$ Find the Sum of the First $n$ Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first $30$ terms of each arithmetic sequence.

1. $7,4,1,-2,-5, \dots$
2. $1,6,11,16,21, \ldots$

Answer

1. $-430$

Exercise $\PageIndex{12}$ Find the Sum of the First $n$ Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.

1. $a_{n}=4 n+7$
2. $a_{n}=-2 n+19$

Answer

1. $585$

Exercise $\PageIndex{13}$ Find the Sum of the First $n$ Terms of an Arithmetic Sequence

In the following exercises, find each sum.

1. $\sum_{i=1}^{50}(4 i-5)$
2. $\sum_{i=1}^{30}(-3 i-7)$
3. $\sum_{i=1}^{35}(i+10)$

Answer

1. $4850$

3. $980$

Geometric Sequences and Series

Exercise $\PageIndex{14}$ Determine if a Sequence is Geometric

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

1. $3,12,48,192,768,3072, \dots$
2. $5,10,15,20,25,30, \dots$
3. $112,56,28,14,7, \frac{7}{2}, \ldots$
4. $9,-18,36,-72,144,-288, \dots$

Answer

2. The sequence is not geometric.

4. The sequence is geometric with common ratio $r=−2$.

Exercise $\PageIndex{15}$ Determine if a Sequence is Geometric

In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.

1. $a_{1}=-3$ and $r=5$
2. $a_{1}=128$ and $r=\frac{1}{4}$
3. $a_{1}=5$ and $r=-3$

Answer

2. $128,32,8,2, \frac{1}{2}$

Exercise $\PageIndex{16}$ Find the General Term ($n$th Term) of a Geometric Sequence

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

1. Find $a_{9}$ given $a_{1}=6$ and $r=2$
2. Find $a_{11}$ given $a_{1}=10,000,000$ and $r=0.1$

Answer

1. $1,536$

Exercise $\PageIndex{17}$ Find the General Term ($n$th Term) of a Geometric Sequence

In the following exercises, find the indicated term of the given sequence. Find the general term of the sequence.

1. Find $a_{12}$ of the sequence, $6,-24,96,-384,1536,-6144, \dots$
2. Find $a_{9}$ of the sequence, $4374,1458,486,162,54,18, \ldots$

Answer

1. $a_{12}=-25,165,824 .$ The general term is $a_{n}=6(-4)^{n-1}$

Exercise $\PageIndex{18}$ Find the Sum of the First $n$ terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

1. $-4,8,-16,32,-64,128 \ldots$
2. $3,12,48,192,768,3072 \ldots$
3. $3125,625,125,25,5,1 \ldots$

Answer

1. $5,460$

3. $\approx 3906.25$

Exercise $\PageIndex{19}$ find the Sum of the First $n$ terms of a Geometric Sequence

In the following exercises, find the sum

1. $\sum_{i=1}^{8} 7(3)^{i}$
2. $\sum_{i=1}^{6} 24\left(\frac{1}{2}\right)^{i}$

Answer

2. $\frac{189}{8}=23.625$

Exercise $\PageIndex{20}$ Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

1. $1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}+\frac{1}{729}-\dots$
2. $49+7+1+\frac{1}{7}+\frac{1}{49}+\frac{1}{343}+\ldots$

Answer

2. $\frac{343}{6} \approx 57.167$

Exercise $\PageIndex{21}$ Find the Sum of an Infinite Geometric Series

In the following exercises, write each repeating decimal as a fraction.

1. $0 . \overline{8}$
2. $0 . \overline{36}$

Answer

2. $\frac{4}{11}$

Exercise $\PageIndex{22}$ Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

1. What is the total effect on the economy of a government tax rebate of $$360$ to each household in order to stimulate the economy if each household will spend $60$% of the rebate in goods and services?
2. Adam just got his first full-time job after graduating from high school at age 17. He decided to invest $$300$ per month in an IRA (an annuity). The interest on the annuity is $7$% which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-seventh birthday?

Answer

2. $\$ 1,634,421.27$

Binomial Theorem

Exercise $\PageIndex{23}$ Use Pascal's Triangle to Expand a Binomial

In the following exercises, expand each binomial using Pascal’s Triangle.

1. $(a+b)^{7}$
2. $(x-y)^{4}$
3. $(x+6)^{3}$
4. $(2 y-3)^{5}$
5. $(7 x+2 y)^{3}$

Answer

2. $x^{4}-4 x^{3} y+6 x^{2} y^{2}-4 x y^{3}+y^{4}$

4. $\begin{array}{l}{32 y^{5}-240 y^{4}+720 y^{3}-1080 y^{2}} {+810 y-243}\end{array}$

Exercise $\PageIndex{24}$ Evaluate a Binomial Coefficient

In the following exercises, evaluate.

1. 1. $\left( \begin{array}{l}{11} \\ {1}\end{array}\right)$
   2. $\left( \begin{array}{l}{12} \\ {12}\end{array}\right)$
   3. $\left( \begin{array}{l}{13} \\ {0}\end{array}\right)$
   4. $\left( \begin{array}{l}{8} \\ {3}\end{array}\right)$
2. 1. $\left( \begin{array}{l}{7} \\ {1}\end{array}\right)$
   2. $\left( \begin{array}{l}{5} \\ {5}\end{array}\right)$
   3. $\left( \begin{array}{l}{9} \\ {0}\end{array}\right)$
   4. $\left( \begin{array}{l}{9} \\ {5}\end{array}\right)$
3. 1. $\left( \begin{array}{l}{1} \\ {1}\end{array}\right)$
   2. $\left( \begin{array}{l}{15} \\ {15}\end{array}\right)$
   3. $\left( \begin{array}{l}{4} \\ {0}\end{array}\right)$
   4. $\left( \begin{array}{l}{11} \\ {2}\end{array}\right)$

Answer

1.

1. $11$
2. $1$
3. $1$
4. $56$

3.

1. $1$
2. $1$
3. $1$
4. $55$

Exercise $\PageIndex{25}$ Use the Binomial Theorem to Expand a Binomial

In the following exercises, expand each binomial, using the Binomial Theorem.

1. $(p+q)^{6}$
2. $(t-1)^{9}$
3. $(2 x+1)^{4}$
4. $(4 x+3 y)^{4}$
5. $(x-3 y)^{5}$

Answer

2. $\begin{array}{l}{t^{9}-9 t^{8}+36 t^{7}-84 t^{6}+126 t^{5}} {-126 t^{4}+84 t^{3}-36 t^{2}+9 t-1}\end{array}$

4. $\begin{array}{l}{256 x^{4}+768 x^{3} y+864 x^{2} y^{2}} {+432 x y^{3}+81 y^{4}}\end{array}$

Exercise $\PageIndex{26}$ Use the Binomial Theorem to Expand a Binomial

In the following exercises, find the indicated term in the expansion of the binomial.

1. Seventh term of $(a+b)^{9}$
2. Third term of $(x-y)^{7}$

Answer

1. $84a^{6} b^{3}$

Exercise $\PageIndex{27}$ Use the Binomial Theorem to Expand a Binomial

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

1. $y^{4}$ term of $(y+3)^{6}$
2. $x^{5}$ term of $(x-2)^{8}$
3. $a^{3} b^{4}$ term of $(2 a+b)^{7}$

Answer

1. $135$

3. $280$

Practice Test

Exercise $\PageIndex{28}$

In the following exercises, write the first five terms of the sequence whose general term is given.

1. $a_{n}=\frac{5 n-3}{3^{n}}$
2. $a_{n}=\frac{(n+2) !}{(n+3) !}$
3. Find a general term for the sequence, $-\frac{2}{3},-\frac{4}{5},-\frac{6}{7},-\frac{8}{9},-\frac{10}{11}, \dots$
4. Expand the partial sum and find its value. $\sum_{i=1}^{4}(-4)^{i}$
5. Write the following using summation notation. $-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}$
6. Write the first five terms of the arithmetic sequence with the given first term and common difference. $a_{1}=-13$ and $d=3$
7. Find the twentieth term of an arithmetic sequence where the first term is two and the common difference is $−7$.
8. Find the twenty-third term of an arithmetic sequence whose seventh term is $11$ and common difference is three. Then find a formula for the general term.
9. Find the first term and common difference of an arithmetic sequence whose ninth term is $−1$ and the sixteenth term is $−15$. Then find a formula for the general term.
10. Find the sum of the first $25$ terms of the arithmetic sequence, $5,9,13,17,21, \dots$
11. Find the sum of the first $50$ terms of the arithmetic sequence whose general term is $a_{n}=-3 n+100$.
12. Find the sum. $\sum_{i=1}^{40}(5 i-21)$

Answer

2. $\frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}$

4. $-4+16-64+256=204$

6. $-13,-10,-7,-4,-1$

8. $a_{23}=59 .$ The general term is $a_{n}=3 n-10$.

10. $1,325$

12. $3,260$

Exercise $\PageIndex{29}$

In the following exercises, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.

1. $14,3,-8,-19,-30,-41, \ldots$
2. $324,108,36,12,4, \frac{4}{3}, \ldots$
3. Write the first five terms of the geometric sequence with the given first term and common ratio. $a_{1}=6$ and $r=−2$.
4. In the geometric sequence whose first term and common ratio are $a_{1}=5$ and $r=4$, find $a_{11}$.
5. Find $a_{10}$ of the geometric sequence, $1250,250,50,10,2, \frac{2}{5}, \ldots$ Then find a
   formula for the general term.
6. Find the sum of the first thirteen terms of the geometric sequence, $2,-6,18,-54,162,-486 \ldots$

Answer

2. The sequence is geometric with common ratio $r=\frac{1}{3}$.

4. $5,242,880$

6. $797,162$

Exercise $\PageIndex{30}$

In the following exercises, find the sum.

1. $\sum_{i=1}^{9} 5(2)^{i}$
2. $1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+\frac{1}{625}-\frac{1}{3125}+\dots$
3. Write the repeating decimal as a fraction. $0 . \overline{81}$
4. Dave just got his first full-time job after graduating from high school at age 18. He decided to invest $$450$ per month in an IRA (an annuity). The interest on the annuity is $6$% which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-fifth birthday?
5. Expand the binomial using Pascal’s Triangle. $(m-2 n)^{5}$
6. Evaluate each binomial coefficient.
   1. $\left( \begin{array}{l}{8} \\ {1}\end{array}\right)$
   2. $\left( \begin{array}{l}{16} \\ {16}\end{array}\right)$
   3. $\left( \begin{array}{l}{12} \\ {0}\end{array}\right)$
   4. $\left( \begin{array}{l}{10} \\ {6}\end{array}\right)$
7. Expand the binomial using the Binomial Theorem. $(4 x+5 y)^{3}$

Answer

2. $\frac{5}{6}$

4. $\$ 1,409,344.19$

6.

1. $8$
2. $1$
3. $1$
4. $210$