12.2E: Exercises
- Page ID
- 30577
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
In the following exercises, write the first five terms of the sequence whose general term is given.
- \(a_{n}=2 n-7\)
- \(a_{n}=5 n-1\)
- \(a_{n}=3 n+1\)
- \(a_{n}=4 n+2\)
- \(a_{n}=2^{n}+3\)
- \(a_{n}=3^{n}-1\)
- \(a_{n}=3^{n}-2 n\)
- \(a_{n}=2^{n}-3 n\)
- \(a_{n}=\frac{2^{n}}{n^{2}}\)
- \(a_{n}=\frac{3^{n}}{n^{3}}\)
- \(a_{n}=\frac{4 n-2}{2^{n}}\)
- \(a_{n}=\frac{3 n+3}{3^{n}}\)
- \(a_{n}=(-1)^{n} \cdot 2 n\)
- \(a_{n}=(-1)^{n} \cdot 3 n\)
- \(a_{n}=(-1)^{n+1} n^{2}\)
- \(a_{n}=(-1)^{n+1} n^{4}\)
- \(a_{n}=\frac{(-1)^{n+1}}{n^{2}}\)
- \(a_{n}=\frac{(-1)^{n+1}}{2 n}\)
- Answer
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1. \(-5,-3,-1,1,3\)
3. \(4,7,10,13,16\)
5. \(5,7,11,19,35\)
7. \(1,5,21,73,233\)
9. \(2,1, \frac{8}{9}, 1, \frac{32}{25}\)
11. \(1, \frac{3}{2}, \frac{5}{4}, \frac{7}{8}, \frac{9}{16}\)
13. \(-2,4,-6,8,-10\)
15. \(1,-4,9,-16,25\)
17. \(1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}\)
In the following exercises, find a general term for the sequence whose first five terms are shown.
- \(8,16,24,32,40, \dots\)
- \(7,14,21,28,35, \ldots\)
- \(6,7,8,9,10, \dots\)
- \(-3,-2,-1,0,1, \dots\)
- \(e^{3}, e^{4}, e^{5}, e^{6}, e^{7}, \ldots\)
- \(\frac{1}{e^{2}}, \frac{1}{e}, 1, e, e^{2}, \ldots\)
- \(-5,10,-15,20,-25, \dots\)
- \(-6,11,-16,21,-26, \dots\)
- \(-1,8,-27,64,-125, \dots\)
- \(2,-5,10,-17,26, \dots\)
- \(-2,4,-6,8,-10, \dots\)
- \(1,-3,5,-7,9, \dots\)
- \(\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1,024}, \dots\)
- \(\frac{1}{1}, \frac{1}{8}, \frac{1}{27}, \frac{1}{64}, \frac{1}{125}, \dots\)
- \(-\frac{1}{2},-\frac{2}{3},-\frac{3}{4},-\frac{4}{5},-\frac{5}{6}, \dots\)
- \(-2,-\frac{3}{2},-\frac{4}{3},-\frac{5}{4},-\frac{6}{5}, \dots\)
- \(-\frac{5}{2},-\frac{5}{4},-\frac{5}{8},-\frac{5}{16},-\frac{5}{32}, \dots\)
- \(4, \frac{1}{2}, \frac{4}{27}, \frac{4}{64}, \frac{4}{125}, \dots\)
- Answer
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1. \(a_{n}=8 n\)
3. \(a_{n}=n+5\)
5. \(a_{n}=e^{n+2}\)
7. \(a_{n}=(-1)^{n} 5 n\)
9. \(a_{n}=(-1)^{n} n^{3}\)
11. \(a_{n}=(-1)^{n} 2 n\)
13. \(a_{n}=\frac{1}{4^{n}}\)
15. \(a_{n}=-\frac{n}{n+1}\)
17. \(-\frac{5}{2^{n}}\)
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
- \(a_{n}=\frac{4}{n !}\)
- \(a_{n}=\frac{5}{n !}\)
- \(a_{n}=3 n !\)
- \(a_{n}=2 n !\)
- \(a_{n}=(2 n) !\)
- \(a_{n}=(3 n) !\)
- \(a_{n}=\frac{(n-1) !}{(n) !}\)
- \(a_{n}=\frac{n !}{(n+1) !}\)
- \(a_{n}=\frac{n !}{n^{2}}\)
- \(a_{n}=\frac{n^{2}}{n !}\)
- \(a_{n}=\frac{(n+1) !}{n^{2}}\)
- \(a_{n}=\frac{(n+1) !}{2 n}\)
- Answer
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1. \(4,2, \frac{2}{3}, \frac{1}{6}, \frac{1}{30}\)
3. \(3,6,18,72,360\)
5. \(2,24,720,40320,3628800\)
7. \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\)
9. \(1, \frac{1}{2}, \frac{2}{3}, \frac{3}{2}, \frac{24}{5}\)
11. \(2, \frac{3}{2}, \frac{8}{3}, \frac{15}{2}, \frac{144}{5}\)
In the following exercises, expand the partial sum and find its value.
- \(\sum_{i=1}^{5} i^{2}\)
- \(\sum_{i=1}^{5} i^{3}\)
- \(\sum_{i=1}^{6}(2 i+3)\)
- \(\sum_{i=1}^{6}(3 i-2)\)
- \(\sum_{i=1}^{4} 2^{i}\)
- \(\sum_{i=1}^{4} 3^{i}\)
- \(\sum_{k=0}^{3} \frac{4}{k !}\)
- \(\sum_{k=0}^{4}-\frac{1}{k !}\)
- \(\sum_{k=1}^{5} k(k+1)\)
- \(\sum_{k=1}^{5} k(2 k-3)\)
- \(\sum_{n=1}^{5} \frac{n}{n+1}\)
- \(\sum_{n=1}^{4} \frac{n}{n+2}\)
- Answer
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1. \(1+4+9+16+25=55\)
3. \(5+7+9+11+13+15=60\)
5. \(2+4+8+16=30\)
7. \(\frac{4}{1}+\frac{4}{1}+\frac{4}{2}+\frac{4}{6}+\frac{32}{3}=10 \frac{2}{3}\)
9. \(2+6+12+20+30=70\)
11. \(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}=\frac{71}{20}\)
In the following exercises, write each sum using summation notation.
- \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}\)
- \(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}\)
- \(1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}\)
- \(\frac{1}{5}+\frac{1}{25}+\frac{1}{125}+\frac{1}{625}\)
- \(2+1+\frac{2}{3}+\frac{1}{2}+\frac{2}{5}\)
- \(3+\frac{3}{2}+1+\frac{3}{4}+\frac{3}{5}+\frac{1}{2}\)
- \(3-6+9-12+15\)
- \(-5+10-15+20-25\)
- \(-2+4-6+8-10+\ldots+20\)
- \(1-3+5-7+9+\ldots+21\)
- \(14+16+18+20+22+24+26\)
- \(9+11+13+15+17+19+21\)
- Answer
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1. \(\sum_{n=1}^{5} \frac{1}{3^{n}}\)
3. \(\sum_{n=1}^{5} \frac{1}{n^{3}}\)
5. \(\sum_{n=1}^{5} \frac{2}{n}\)
7. \(\sum_{n=1}^{5}(-1)^{n+1} 3 n\)
9. \(\sum_{n=1}^{10}(-1)^{n} 2 n\)
11. \(\sum_{n=1}^{7}(2 n+12)\)
- In your own words, explain how to write the terms of a sequence when you know the formula. Show an example to illustrate your explanation.
- Which terms of the sequence are negative when the \(n^{th}\) term of the sequence is \(a_{n}=(-1)^{n}(n+2)\)?
- In your own words, explain what is meant by \(n!\) Show some examples to illustrate your explanation.
- Explain what each part of the notation \(\sum_{k=1}^{12} 2 k\) means.
- Answer
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1. Answers will vary.
3. Answers will vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.