Chapter 12 Review Exercises
Sequences
In the following exercises, write the first five terms of the sequence whose general term is given.
- \(a_{n}=7 n-5\)
- \(a_{n}=3^{n}+4\)
- \(a_{n}=2^{n}+n\)
- \(a_{n}=\frac{2 n+1}{4^{n}}\)
- \(a_{n}=\frac{(-1)^{n}}{n^{2}}\)
- Answer
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2. \(7,13,31,85,247\)
4. \(\frac{3}{4}, \frac{5}{16}, \frac{7}{64}, \frac{9}{256}, \frac{11}{1024}\)
In the following exercises, find a general term for the sequence whose first five terms are shown.
- \(9,18,27,36,45, \dots\)
- \(-5,-4,-3,-2,-1, \dots\)
- \(\frac{1}{e^{3}}, \frac{1}{e^{2}}, \frac{1}{e}, 1, e, \ldots\)
- \(1,-8,27,-64,125, \ldots\)
- \(-\frac{1}{3},-\frac{1}{2},-\frac{3}{5},-\frac{2}{3},-\frac{5}{7}, \dots\)
- Answer
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1. \(a_{n}=9 n\)
3. \(a_{n}=e^{n-4}\)
5. \(a_{n}=-\frac{n}{n+2}\)
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
- \(a_{n}=4 n !\)
- \(a_{n}=\frac{n !}{(n+2) !}\)
- \(a_{n}=\frac{(n-1) !}{(n+1)^{2}}\)
- Answer
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2. \(\frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, \frac{1}{42}\)
In the following exercises, expand the partial sum and find its value.
- \(\sum_{i=1}^{7}(2 i-5)\)
- \(\sum_{i=1}^{3} 5^{i}\)
- \(\sum_{k=0}^{4} \frac{4}{k !}\)
- \(\sum_{k=1}^{4}(k+1)(2 k+1)\)
- Answer
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1. \(\begin{array}{l}{-3+(-1)+1+3+5} {+7+9=21}\end{array}\)
3. \(4+4+2+\frac{2}{3}+\frac{1}{6}=\frac{65}{6}\)
In the following exercises, write each sum using summation notation.
- \(-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}\)
- \(4-8+12-16+20-24\)
- \(4+2+\frac{4}{3}+1+\frac{4}{5}\)
- Answer
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1. \(\sum_{n=1}^{5}(-1)^{n} \frac{1}{3^{n}}\)
3. \(\sum_{n=1}^{5} \frac{4}{n}\)
Arithmetic Sequences
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
- \(1,2,4,8,16,32, \dots\)
- \(-7,-1,5,11,17,23, \dots\)
- \(13,9,5,1,-3,-7, \dots\)
- Answer
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2. The sequence is arithmetic with common difference \(d=6\).
In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.
- \(a_{1}=5\) and \(d=3\)
- \(a_{1}=8\) and \(d=-2\)
- \(a_{1}=-13\) and \(d=6\)
- Answer
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1. \(5,8,11,14,17\)
3. \(-13,-7,-1,5,11\)
In the following exercises, find the term described using the information provided.
- Find the twenty-fifth term of a sequence where the first term is five and the common difference is three.
- Find the thirtieth term of a sequence where the first term is \(16\) and the common difference is \(−5\).
- Find the seventeenth term of a sequence where the first term is \(−21\) and the common difference is two.
- Answer
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2. \(-129\)
In the following exercises, find the indicated term and give the formula for the general term.
- Find the eighteenth term of a sequence where the fifth term is \(12\) and the common difference is seven.
- Find the twenty-first term of a sequence where the seventh term is \(14\) and the common difference is \(−3\).
- Answer
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1. \(a_{18}=103 .\) The general term is \(a_{n}=7 n-23\).
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.
- The fifth term is \(17\) and the fourteenth term is \(53\).
- The third term is \(−26\) and the sixteenth term is \(−91\).
- Answer
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1. \(a_{1}=1, d=4 .\) The general term is \(a_{n}=4 n-3\).
In the following exercises, find the sum of the first \(30\) terms of each arithmetic sequence.
- \(7,4,1,-2,-5, \dots\)
- \(1,6,11,16,21, \ldots\)
- Answer
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1. \(-430\)
In the following exercises, find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.
- \(a_{n}=4 n+7\)
- \(a_{n}=-2 n+19\)
- Answer
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1. \(585\)
In the following exercises, find each sum.
- \(\sum_{i=1}^{50}(4 i-5)\)
- \(\sum_{i=1}^{30}(-3 i-7)\)
- \(\sum_{i=1}^{35}(i+10)\)
- Answer
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1. \(4850\)
3. \(980\)
Geometric Sequences and Series
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
- \(3,12,48,192,768,3072, \dots\)
- \(5,10,15,20,25,30, \dots\)
- \(112,56,28,14,7, \frac{7}{2}, \ldots\)
- \(9,-18,36,-72,144,-288, \dots\)
- Answer
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2. The sequence is not geometric.
4. The sequence is geometric with common ratio \(r=−2\).
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
- \(a_{1}=-3\) and \(r=5\)
- \(a_{1}=128\) and \(r=\frac{1}{4}\)
- \(a_{1}=5\) and \(r=-3\)
- Answer
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2. \(128,32,8,2, \frac{1}{2}\)
In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.
- Find \(a_{9}\) given \(a_{1}=6\) and \(r=2\)
- Find \(a_{11}\) given \(a_{1}=10,000,000\) and \(r=0.1\)
- Answer
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1. \(1,536\)
In the following exercises, find the indicated term of the given sequence. Find the general term of the sequence.
- Find \(a_{12}\) of the sequence, \(6,-24,96,-384,1536,-6144, \dots\)
- Find \(a_{9}\) of the sequence, \(4374,1458,486,162,54,18, \ldots\)
- Answer
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1. \(a_{12}=-25,165,824 .\) The general term is \(a_{n}=6(-4)^{n-1}\)
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
- \(-4,8,-16,32,-64,128 \ldots\)
- \(3,12,48,192,768,3072 \ldots\)
- \(3125,625,125,25,5,1 \ldots\)
- Answer
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1. \(5,460\)
3. \(\approx 3906.25\)
In the following exercises, find the sum
- \(\sum_{i=1}^{8} 7(3)^{i}\)
- \(\sum_{i=1}^{6} 24\left(\frac{1}{2}\right)^{i}\)
- Answer
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2. \(\frac{189}{8}=23.625\)
In the following exercises, find the sum of each infinite geometric series.
- \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}+\frac{1}{729}-\dots\)
- \(49+7+1+\frac{1}{7}+\frac{1}{49}+\frac{1}{343}+\ldots\)
- Answer
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2. \(\frac{343}{6} \approx 57.167\)
In the following exercises, write each repeating decimal as a fraction.
- \(0 . \overline{8}\)
- \(0 . \overline{36}\)
- Answer
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2. \(\frac{4}{11}\)
In the following exercises, solve the problem.
- 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?
- 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
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2. \(\$ 1,634,421.27\)
Binomial Theorem
In the following exercises, expand each binomial using Pascal’s Triangle.
- \((a+b)^{7}\)
- \((x-y)^{4}\)
- \((x+6)^{3}\)
- \((2 y-3)^{5}\)
- \((7 x+2 y)^{3}\)
- Answer
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2. \(x^{4}-4 x^{3} y+6 x^{2} y^{2}-4 x y^{3}+y^{4}\)
4. \(\begin{array}{l}{32 y^{5}-240 y^{4}+720 y^{3}-1080 y^{2}} {+810 y-243}\end{array}\)
In the following exercises, evaluate.
-
- \(\left( \begin{array}{l}{11} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{12} \\ {12}\end{array}\right)\)
- \(\left( \begin{array}{l}{13} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{8} \\ {3}\end{array}\right)\)
-
- \(\left( \begin{array}{l}{7} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{5} \\ {5}\end{array}\right)\)
- \(\left( \begin{array}{l}{9} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{9} \\ {5}\end{array}\right)\)
-
- \(\left( \begin{array}{l}{1} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{15} \\ {15}\end{array}\right)\)
- \(\left( \begin{array}{l}{4} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{11} \\ {2}\end{array}\right)\)
- Answer
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1.
- \(11\)
- \(1\)
- \(1\)
- \(56\)
3.
- \(1\)
- \(1\)
- \(1\)
- \(55\)
In the following exercises, expand each binomial, using the Binomial Theorem.
- \((p+q)^{6}\)
- \((t-1)^{9}\)
- \((2 x+1)^{4}\)
- \((4 x+3 y)^{4}\)
- \((x-3 y)^{5}\)
- Answer
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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. \(\begin{array}{l}{256 x^{4}+768 x^{3} y+864 x^{2} y^{2}} {+432 x y^{3}+81 y^{4}}\end{array}\)
In the following exercises, find the indicated term in the expansion of the binomial.
- Seventh term of \((a+b)^{9}\)
- Third term of \((x-y)^{7}\)
- Answer
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1. \(84a^{6} b^{3}\)
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.
- \(y^{4}\) term of \((y+3)^{6}\)
- \(x^{5}\) term of \((x-2)^{8}\)
- \(a^{3} b^{4}\) term of \((2 a+b)^{7}\)
- Answer
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1. \(135\)
3. \(280\)
Practice Test
In the following exercises, write the first five terms of the sequence whose general term is given.
- \(a_{n}=\frac{5 n-3}{3^{n}}\)
- \(a_{n}=\frac{(n+2) !}{(n+3) !}\)
- Find a general term for the sequence, \(-\frac{2}{3},-\frac{4}{5},-\frac{6}{7},-\frac{8}{9},-\frac{10}{11}, \dots\)
- Expand the partial sum and find its value. \(\sum_{i=1}^{4}(-4)^{i}\)
- Write the following using summation notation. \(-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}\)
- Write the first five terms of the arithmetic sequence with the given first term and common difference. \(a_{1}=-13\) and \(d=3\)
- Find the twentieth term of an arithmetic sequence where the first term is two and the common difference is \(−7\).
- 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.
- 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.
- Find the sum of the first \(25\) terms of the arithmetic sequence, \(5,9,13,17,21, \dots\)
- Find the sum of the first \(50\) terms of the arithmetic sequence whose general term is \(a_{n}=-3 n+100\).
- Find the sum. \(\sum_{i=1}^{40}(5 i-21)\)
- Answer
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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. \(3,260\)
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.
- \(14,3,-8,-19,-30,-41, \ldots\)
- \(324,108,36,12,4, \frac{4}{3}, \ldots\)
- Write the first five terms of the geometric sequence with the given first term and common ratio. \(a_{1}=6\) and \(r=−2\).
- In the geometric sequence whose first term and common ratio are \(a_{1}=5\) and \(r=4\), find \(a_{11}\).
-
Find \(a_{10}\) of the geometric sequence, \(1250,250,50,10,2, \frac{2}{5}, \ldots\) Then find a
formula for the general term. - Find the sum of the first thirteen terms of the geometric sequence, \(2,-6,18,-54,162,-486 \ldots\)
- Answer
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2. The sequence is geometric with common ratio \(r=\frac{1}{3}\).
4. \(5,242,880\)
6. \(797,162\)
In the following exercises, find the sum.
- \(\sum_{i=1}^{9} 5(2)^{i}\)
- \(1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+\frac{1}{625}-\frac{1}{3125}+\dots\)
- Write the repeating decimal as a fraction. \(0 . \overline{81}\)
- 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?
- Expand the binomial using Pascal’s Triangle. \((m-2 n)^{5}\)
-
Evaluate each binomial coefficient.
- \(\left( \begin{array}{l}{8} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{16} \\ {16}\end{array}\right)\)
- \(\left( \begin{array}{l}{12} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{10} \\ {6}\end{array}\right)\)
- Expand the binomial using the Binomial Theorem. \((4 x+5 y)^{3}\)
- Answer
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2. \(\frac{5}{6}\)
4. \(\$ 1,409,344.19\)
6.
- \(8\)
- \(1\)
- \(1\)
- \(210\)