Proof That The Sum The First N Odd Integers is a Square

The sum of the first n odd integers is a square. This is a theorem and can easily be proven if you have already learned proof by mathematical induction. Even though it sounds like a boring theorem, it is actually very interesting if represented visually. Consider the following diagrams and their numerical representations.

$latex 1 + 3 = 2^2$

$latex 1 + 3 + 5 = 3^2$

Can you see the pattern? Let’s have some more examples.  Continue reading

Representing the Sum and Difference of Two Squares

One of the most beautiful characteristics of mathematics is you can create multiple representations of a single concept. Functions for example can represented in words, equations, graphs, or tables.

In this post, we represent geometrically an algebraic concept namely the sum and difference of two squares. The difference of two squares states you can factor $latex a^2 – b^2$ into $latex (a+b)(a-b)$. You may use a piece of paper to follow the procedure below.

Geometric Representation

1. Take a square with length $latex a$ (any convenient side length)

2. Cut out a small square with side length $latex b$ from one of the corners of the large square. The area of the remaining figure is $latex a^2 – b^2$. Can you see why? Continue reading

A Geometric Proof of an Infinite Series

In the previous post, we have seen how easy it is to prove a problem in Geometry using Algebra. The problem could also be proven geometrically, but the proof is longer. In this post, we will learn how to use Geometry to prove a problem on infinite series. That is, we have to show geometrically, the sum of

$latex \displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots$.

Note that the symbol $latex \cdots$ means that the number of terms is infinite; that is, the addition continues without end.  In this post, we show that the sum of this infinite series is 1.

Theorem

$latex \displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots = 1$