# Examples Using Mathematical Induction

by Dave
(DAVE)—

In this video, I’m going to work through five examples using mathematical induction. The first two examples involve summations, and the last three examples have an inequality. In each of these examples, I work through specials cases before wiring a proof so that you can get a feel for the result before actually proving the result holds using mathematical induction.

Here are four examples using mathematical induction. These examples assume that you have little or no knowledge of mathematical induction. In addition, in these examples, I show you the scratch involved in finding proof.

## Here are 5 Examples Using Mathematical Induction

In the first example, we work through four special cases. These specials cases are helpful if you are not that familiar with summations, for instance.

Example. Prove that for all positive integers $n$, $$\sum_{i=1}^n (2i-1)=n^2.$$ Example 1: Using mathematical induction to prove that summation of odds equals perfect square.

Example. In the following, use mathematical induction to show that $$\sum _{k=1}^n (-1)^k k = \frac{(-1)^n(2n+1)-1}{4}$$ for every positive integer $n.$

Example. In the following, use mathematical induction to prove that $3^n>3n-1$ for every positive integer $n.$

Example. Let $x$ be any real number greater than $-1.$ Use mathematical induction to prove that $(1+x)^n\geq 1+n x$ for all positive integers $n.$

Example. In the following, use mathematical induction to prove the inequality. For all $n\geq 1$, $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{n^2}\leq 2-\frac{1}{n}.$$