# Exercises on Mathematical Induction (10 Must See)

by Dave
(DAVE)—

What are the four types of induction problems that are must-see for any student just starting out learning mathematical induction? In this video, I work through 10 exercises using mathematical induction. For each of these exercises, I demonstrate how to brainstorm the problem and then how to write up a rigorous solution. Some of these exercises involve the factorial function, and others involve the well-ordering of the natural numbers.

First, I’m going to mention the four types of problems I am interested in for this video. These are the main types of exercises on mathematical induction a student will see before learning more advanced mathematics.

• addition and multiplication
• well-ordering
• factorial
• real world

Of course, mathematical induction is ubiquitous in mathematics, meaning mathematical induction is used throughout mathematics and is a fundamental proof technique. As you learn more mathematics, you will see instantiations of mathematical induction.

## 4 exercises on mathematical induction that only involve addition and multiplication

Exercise. Prove that, for all positive integers $n$, $$\sum _{k=1}^n k^3=\left(\frac{n(n+1)}{2}\right)^2.$$

Exercise. Prove that, for all positive integers $n$, $$\sum _{j=1}^n (-1)^{j-1}j^2=(-1)^{n-1}\frac{n(n+1)}{2}.$$

Exercise. Use mathematical induction to prove that $$\sum_{j=1}^n j(j+1) = \frac{n(n+1)(n+2)}{3}$$ for all natural numbers $n\geq 1.$

Exercise. In the following, use the strong form of mathematical induction to establish that for all $n\geq 1,$ $$a^n-1=(a-1)\left(a^{n-1}+a^{n-2}+a^{n-3}+\cdots+a+1\right).$$

## 3 exercises that involve the well-ordering on the natural numbers

To see the remaining solutions for these problems check out the video.

Exercise Show that $2^{n+1} > n+2$ for every positive integer $n$.

Exercise. Use mathematical induction to prove that $2^n > n^2$ for $n>4.$

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

## 2 exercises on mathematical induction that involve the factorial

These induction exercises involve the factorial function which makes them fun to do.

Exercise. Use mathematical induction to show that $$2\cdot 6\cdot 10\cdot 14\cdots (4n-2)=\frac{(2n)!}{n!}$$ for every positive integer $n.$

Exercise Use mathematical induction to prove that $$(2n)! < 2^{2n}(n!)^2$$ for all natural numbers $n\geq 1.$

## 1 exercise on mathematical induction that involves the real world (explicitly)

Solutions to the following problem sometimes involve strong induction. Our proof does not. Check out the video to see how we solve this problem with regular induction.

Exercise. Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps.

## Conclusion

And there they are, the ten must-see exercises on mathematical induction that every student who is just learning mathematical induction should see. If you like to see the complete guide to mathematical induction, in this guide, I take you through the process from beginner to mastering mathematical induction.