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**. 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 $n.$

This last example was especially fun to work through.

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

## 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, itâ€™s usually is relatively easy to reproduce unless youâ€™re working through a tricky problem. In conclusion, in an upcoming video, Iâ€™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.