4.S: Mathematical Induction (Summary)
- Page ID
- 7058
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Important Definitions
- Inductive set, page 171
- Factorial, page 201
- Recursive definition, page 200
- Fibonacci numbers, page 202
- Geometric sequence, page 206
- Geometric series, page 206
The Various Forms of Mathematical Induction
- The Principle of Mathematical Induction
If \(T\) is a subset of \(\mathbb{N}\) such that
(a) \(1 \in T\), and
(b) For every \(k \in \mathbb{N}\), if \(k \in T\), then \((k + 1) \in T\).
then \(T = \mathbb{N}\)
Procedure for a Proof by Mathematical Induction
To prove \((\forall n \in \mathbb{N}\)(P(n))\)
\[\begin{array} {rcl} {\text{Basis step}} &: & {\text{Prove} P(1).} \\ {\text{Inductive step}} &: & {\text{Prove that for each} k \in \mathbb{N}, \text{if} P(k) \text{is true, then} P(k + 1) \text{is true.}} \end{array}\] - The Extended Principle of Mathematical Induction
Let \(M\) be an integer. If \(T\) is a subset of \(\mathbb{Z}\) such that
(a) \(M \in T\), and
(b) For every \(k \in \mathbb{Z}\) with \(k \ge M\), if \(k \in T\), then \((k + 1) \in T\).
then \(T\) contains all integers greater than or equal to \(M\).
Using the Extended Principle of Mathematical Induction
Let \(M\) be an intteger. To prove \((\forall n \in \mathbb{Z} \text{with} n \ge M) (P(n))\)
\[\begin{array} {rcl} {\text{Basis step}} &: & {\text{Prove} P(M).} \\ {\text{Inductive step}} &: & {\text{Prove that for each} k \in \mathbb{Z} \text{with} k \ge M, \text{if} P(k) \text{is true, then} P(k + 1) \text{is true.}} \end{array}\]
We can then conclude that \(P(n)\) is true for all \(n \in \mathbb{Z}\) with \(n \ge M\). - The Second Principle of Mathematical Induction
Let \(M\) be an integer. If \(T\) is a subset of \(\mathbb{Z}\) such that
(a) \(M \in T\), and
(b) For every \(k \in \mathbb{Z}\) with \(k \ge M\), if \(\{M, M + 1, ..., k\} \subseteq T\), then \((k + 1) \in T\).
then \(T\) contains all integers greater than or equal to \(M\).
Using the Second Principle of Mathematical Induction
Let \(M\) be an intteger. To prove \((\forall n \in \mathbb{Z} \text{with} n \ge M) (P(n))\)
\[\begin{array} {rcl} {\text{Basis step}} &: & {\text{Prove} P(M).} \\ {\text{Inductive step}} &: & {\text{Let} k \in \mathbb{Z} \text{with} k \ge M. \text{Prove that if} P(M), P(M + 1), ..., P(k) \text{are true, then} P(k + 1) \text{is true.}} \end{array}\]
We can then conclude that \(P(n)\) is true for all \(n \in \mathbb{Z}\) with \(n \ge M\).
Important Results
- Theorem 4.9. Each natural number greater than 1 is either a prime number or is a product of prime numbers.
- Theorem 4.14. Let \(a, r \in \mathbb{R}\). If a geometric sequence is defined by \(a_1 = a\) and for each \(n \in \mathbb{N}\), \(a_{n + 1} = r \cdot a_n\), then for each \(n \in \mathbb{N}\), \(a_n = a \cdot r^{n - 1}\).
- Theorem 4.15. Let \(a, r \in \mathbb{R}\). If the sequence \(S_1, S_2, ..., S_n, ...\) is defined by \(S_1 = a\) and for each \(n \in \mathbb{N}\), \(S_{n + 1} = a + r \cdot S_n\), then for each \(n \in \mathbb{N}\), \(S_n = a + a \cdot r + a \cdot r^2 + \cdot\cdot\cdot + a \cdot r^{n - 1}\). That is, the geometric series \(S_n\) is the sum of the first n terms of the corresponding geometric sequence.
- Theorem 4.16. Let \(a, r \in \mathbb{R}\) and \(r \ne 1\). If the sequence \(S_1, S_2, ..., S_n, ...\) is defined by \(S_1 = a\) and for each \(n \in \mathbb{N}\), \(S_{n + 1} = a + r \cdot S_n\), then for each \(n \in \mathbb{N}\), \(S_n = a (\dfrac{1 - r^n}{1 - r})\).