12.C.E: Exercises for Mathematical Induction
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Prove the given statement by induction for all \(n \geq 1\):
\(1+3+5+7+\ldots+(2 n-1)=n^2\)
Prove the given statement by induction for all \(n \geq 1\):
\(1^2+2^2+\cdots+n^2=\frac{1}{6} n(n+1)(2 n+1)\)
Prove the given statement by induction for all \(n \geq 1\):
\(1^3+2^3+\ldots+n^3=(1+2+\ldots+n)^2\)
Prove the given statement by induction for all \(n \geq 1\):
\(1 \cdot 2+2 \cdot 3+\cdots+n(n+1)= \frac{1}{3} n(n+1)(n+2)\)
Prove the given statement by induction for all \(n \geq 1\):
\(1 \cdot 2^2+2 \cdot 3^2+\cdots+n(n+1)^2= \frac{1}{12} n(n+1)(n+2)(3 n+5)\)
Prove the given statement by induction for all \(n \geq 1\):
\(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}\)
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\(\frac{n}{n+1}+\frac{1}{(n+1)(n+2)}=\frac{n(n+2)+1}{(n+2} =\frac{(n+1)(n+2)}{(n+1)(n+2)}\)
Prove the given statement by induction for all \(n \geq 1\):
\(1^2+3^2+\cdots+(2 n-1)^2=\frac{n}{3} (4 n^2-1)\)
Prove the given statement by induction for all \(n \geq 1\):
\( \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\cdots+\frac{1}{n(n+3)} = \frac{n(n+3)}{4(n+1)(n+2)}\)
Prove the given statement by induction for all \(n \geq 1\):
\(1+2+2^2+\ldots+2^{n-1}=2^n-1\)
Prove the given statement by induction for all \(n \geq 1\):
\(3+3^3+3^5+\cdots+3^{2 n-1}=\frac{3}{8}(9^n-1\)
Prove the given statement by induction for all \(n \geq 1\):
\(\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2} \leq 2-\frac{1}{n}\)
Prove the given statement by induction for all \(n \geq 1\):
\(n<2^n\)
Prove the given statement by induction for all \(n \geq 1\):
For any integer \(m>0\), \(m!n!<(m +n)!\)
Prove the given statement by induction for all \(n \geq 1\):
\[\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \leq 2\sqrt{n}-1 \nonumber\]
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\(2\sqrt{n} - 1 + \frac{1}{\sqrt{n+1}} = \frac{2\sqrt{n^2+n}+1}{\sqrt{n+1}} - 1 < \frac{2(n+1)}{\sqrt{n+1}}-1 = 2\sqrt{n+1}-1\)
Prove the given statement by induction for all \(n \geq 1\):
\(\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \geq \sqrt{n}\)
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Prove the given statement by induction for all \(n \geq 1\):
\(n^{3} + (n + 1)^{3} + (n + 2)^{3}\) is a multiple of \(9\).
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Prove the given statement by induction for all \(n \geq 1\):
\(5n + 3\) is a multiple of \(4\).
Prove the given statement by induction for all \(n \geq 1\):
\(n^{3} - n\) is a multiple of \(3\).
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\(n^{3} - n\) is a multiple of \(3\). \begin{sol} If \(n^{3} -n = 3k\), then \((n + 1)^{3} - (n + 1) = 3k + 3n^{2} + 3n = 3(k + n^{2} + n)\)
Prove the given statement by induction for all \(n \geq 1\):
\(3^{2n+1} + 2^{n+2}\) is a multiple of \(7\).
Let \(B_{n} = 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \dots + n \cdot n\)! Find a formula for \(B_{n}\) and prove it.
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\(B_{n} = (n + 1)! - 1\)
\[ A_n = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})\cdots (1-\frac{1}{n}) \nonumber \]
Let \(A_n=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)(1- \left.\frac{1}{4}\right) \cdots\left(1-\frac{1}{n}\right)\)
Find a formula for \(A_n\) and prove it.
Suppose \(S_{n}\) is a statement about \(n\) for each \(n \geq 1\). Explain what must be done to prove that \(S_{n}\) is true for all \(n \geq 1\) if it is known that:
- \(S_{n} \Rightarrow S_{n+2}\) for each \(n \geq 1\).
- \(S_{n} \Rightarrow S_{n+8}\) for each \(n \geq 1\).
- \(S_{n} \Rightarrow S_{n+1}\) for each \(n \geq 10\).
- Both \(S_{n}\) and \(S_{n+1} \Rightarrow S_{n+2}\) for each \(n \geq 1\).
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- Verify each of \(S_1\), \(S_2\), \(\ldots\), \(S_8\).
If \(S_{n}\) is a statement for each \(n \geq 1\), argue that \(S_{n}\) is true for all \(n \geq 1\) if it is known that the following two conditions hold:
- \(S_{n} \Rightarrow S_{n-1}\) for each \(n \geq 2\).
- \(S_{n}\) is true for infinitely many values of \(n\).
Suppose a sequence \(a_{1}, a_{2}, \dots\) of numbers is given that satisfies:
- \(a_{1} = 2\). \
- \(a_{n+1} = 2a_{n}\) for each \(n \geq 1\).
Formulate a theorem giving \(a_{n}\) in terms of \(n\), and prove your result by induction.
Suppose a sequence \(a_{1}, a_{2}, \dots\) of numbers is given that satisfies:
- \item \(a_{1} = b\).
- \item \(a_{n+1} = ca_{n} + b\) for \(n = 1, 2, 3, \dots\).
Formulate a theorem giving \(a_n\) in terms of \(n\), and prove your result by induction.
- Show that \(n^{2} \leq 2^{n}\) for all \(n \geq 4\).
- Show that \(n^{3} \leq 2^{n}\) for all \(n \geq 10\).