Problems

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 a visual proof, show that \(1^3+2^3+3^3+\cdots+n^3=\frac{1}{4}\left(n(n+1)\right)^2\)

Give a visual proof that the sum of consecutive numbers until \(n\), i.e: \(1+2+\cdots + n\), where \(n\) is some whole number; is equal to \(n(n+1)/2\).

Use a visual proof to find the value of \[\frac{1+3+5+\cdots +2n-1}{(2n+1)+(2n+3)+\cdots + (4n-1)}\] You are not allowed to use the result from the examples to simplify the fraction.

The Pythagorean Theorem is a very useful tool in geometry. It says that if you have a right-angled triangle with sides measuring \(a,b,c\) where \(c\) is the longest side of the three, then \(a^2+b^2=c^2\). Explain how the following diagram gives a visual proof of the Pythagoraen Theorem. You can take it as a fact that the angles inside a triangle add up to \(180^\circ\).

Without carrying out the multiplications, which is larger: \[(2015+2026)^2\qquad \text{or} \qquad 4\times 2015\times 2016\]

Draw a table with \(n+1\) columns and \(n\) rows, such that each column contains the numbers \(1,2,3,\cdots, n\). Explain how this table can be used to give a visual proof of the following identity \[(1^1\times 1!)\times (2^2\times 2!)\times (3^3\times 3!)\times\cdots \times (n^n\times n!)=(n!)^{n+1}\]

By cleverly dividing a square of side length \(1\), show that the sum \[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\cdots + \frac{1}{1024}= \frac{1023}{1024}\]

For natural numbers \(n\) and \(k\) with \(k\leq n\), the notation \({n\choose k}\) means the number of ways one can choose \(k\) objects from a set of \(n\) objects when the ordering does not matter. Explain how the diagram below gives a visual proof of the fact that \[{n+1\choose 2}={n\choose 2}+n.\]

For a real number \(x\), we call \(|x|\) its absolute value. It is defined as whichever is larger: \(x\) or \(-x\). For example, \(|-2|=2\) and \(|3|=3\).

One of the most important inequalities involving absolute values is the triangle inequality, which states that \[|a+b| \le |a| + |b|.\]

Show that this inequality is true.