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How to Prove It: A Structured Approach

Daniel J. Velleman

Chapter 6

Mathematical Induction - all with Video Answers

Educators

Section 1

Proof by mathematical induction

01:02

Problem 1

Prove that for all $n \in \mathbb{N}, 0+1+2+\cdots+n=n(n+1) / 2$.

Linh Vu
Linh Vu
Numerade Educator

Problem 2

Prove that for all $n \in \mathbb{N}, 0^2+1^2+2^2+\cdots+n^2=n(n+1)(2 n+1) / 6$.

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Problem 3

Prove that for all $n \in \mathbb{N}, 0^3+1^3+2^3+\cdots+n^3=[n(n+1) / 2]^2$.

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02:18

Problem 4

Find a formula for $1+3+5+\cdots+(2 n-1)$, for $n \geq 1$, and prove that your formula is correct. (Hint: First try some particular values of $n$ and look for a pattern.)

Tanishq Gupta
Tanishq Gupta
Numerade Educator
04:34

Problem 5

Find a formula for $3^0+3^1+3^2+\cdots+3^n$, for $n \geq 0$, and prove that your formula is correct. (Hint: Try to guess the formula, basing your guess on
Example 6.1.1. Then try out some values of $n$ and adjust your guess if necessary.)

Kevin Harmer
Kevin Harmer
Numerade Educator
10:24

Problem 6

(a) Prove that for all $n \in \mathbb{N}, 2 \mid\left(n^2+n\right)$.
(b) Prove that for all $n \in \mathbb{N}, 6 \mid\left(n^3-n\right)$.

Bobby Barnes
Bobby Barnes
University of North Texas
02:38

Problem 7

Prove that for all $n \in \mathbb{N}, 64 \mid\left(9^n-8 n-1\right)$.

Vg
Viraj Gaggar
Numerade Educator
02:34

Problem 8

Prove that for all integers $a$ and $b$ and all $n \in \mathbb{N},(a-b) \mid\left(a^n-b^n\right)$. (Hint: Let $a$ and $b$ be arbitrary integers and then prove by induction that $\forall n \in \mathbb{N}\left[(a-b) \mid\left(a^n-b^n\right)\right]$. For the induction step, you must relate $a^{n+1}-b^{n+1}$ to $a^n-b^n$. You might find it useful to start by completing the following equation: $a^{n+1}-b^{n+1}=a\left(a^n-b^n\right)+\underline{?}$.)

Tanishq Gupta
Tanishq Gupta
Numerade Educator
08:30

Problem 9

Prove that for all $n \geq 10,2^n>n^3$.

Bobby Barnes
Bobby Barnes
University of North Texas
01:46

Problem 10

10. Suppose $a$ and $b$ are real numbers and $0 \leq a \leq b$.
(a) Prove that for all $n \in \mathbb{N}, a^n \leq b^n$.
(b) Prove that for all $n \in \mathbb{N}, a b^n+b a^n \leq a^{n+1}+b^{n+1}$.
(c) Prove that for all $n \in \mathbb{N}$,
$$
\left(\frac{a+b}{2}\right)^n \leq \frac{a^n+b^n}{2} .
$$

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator