Problems

Explain how you can use the diagram below to show that \(a^2-b^2=(a+b)(a-b)\)

The Arithmetic-Geometric inequality is one of the most famous inequalities. It says that for positive numbers \(a\) and \(b\), \(\frac{a+b}2\geq \sqrt{ab}\). Show this inequality using the diagram below:

In the examples we showed that the sum of consecutive odd numbers starting from one was a perfect square. Now show how the following diagram can be used to give an alternative proof.

Using the following diagram, show that \(1^3+2^3+3^3+\cdots+n^3=\frac{1}{4}\left(n(n+1)\right)^2\)