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 $n$, $$ \sum_{i=1}^n (2i-1)=n^2. $$

examples using mathematical induction summation odds equals perfect square
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 using mathematical induction involving an alternating summation
Mathematical induction involving an alternating summation.

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

Example using mathematical induction involving an inequality
Example using mathematical induction involving an inequality

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 $n.$

Inequality involving a binomial expansion
Inequality involving a binomial expansion

This last example was especially fun to work through.

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

Inequality and summation
Inequality and summation

Conclusion

In each of the examples above, I tried to show you what I was thinking while working through them. After that, you notice that induction can take several practice examples to get working right. But once you get it, it’s usually is relatively easy to reproduce unless you’re working through a tricky problem. In conclusion, in an upcoming video, I’ll discuss other induction principles and different forms of induction, including strong induction.

If you are looking for first examples using mathematical induction check out this video here. To learn a great deal more on this topic, consider taking the online course The Natural Numbers.