# A Beautiful Proof

William John Holden

2014-01-28

- Introduction
- Method One
- Method Two
- A Beautiful Proof

## Introduction

My wife recently noticed this brainteaser online:

We immediately noticed two different and valid methods to calculate the next value.

## Method One

After looking at the problem for a few seconds I noticed that each value on the right is equal to the left value times the sum of the left value and one.

This means the next value in the sequence is:

## Method Two

My wife noticed a different pattern. Each value increments by two more than the previous value. That is:

I suppose accountants read up-and-down, and programmers read right-to-left!

Infer the first value pair:

We can express this series as:

Using the sum identity, we discover

We noticed two different rules to reach the same answer, yet there is an even simpler means of finding the answer: a picture.

## A Beautiful Proof

This is, in my opinion, a beautiful proof. Hopefully next time I can resist thinking so deeply about trending images from Facebook!