# Ken Ward's Mathematics Pages

Series Contents

## Introduction

The English Mathematician, Sir Isaac Newton (1642 - 1727) presented the Binomial Theorem in all its glory without any proof. This was in the mid 1660's. The Swiss Mathematician, Jaques Bernoulli (Jakob Bernoulli) (1654 - 1705) proved the theorem by induction for nonnegative integers. Leonhart Euler (1707 - 1783), also Swiss, presented an algebraic proof for all values of n (which some claim is faulty, but this is arguable). However, Euler proved the conditions under which the Binomial Theorem converges. The Scot, Colin Maclaurin (1698 - 1746) presented a proof in calculus (which was objected to because it was calculus and not algebra). The Norwegian Mathematician, Niels Henrik Abel (1802-1829 ) finally proved the theorem for all values.

We have already proved the Binomial Theorem for nonnegative integers (not using induction), and we have proved it for all values using calculus. Here we seek to prove the theorem for nonnegative integers, this time using mathematical induction.

## Proof By Induction for Nonnegative n

The essence of this proof is to use the addition formula, which we have proved for all real numbers without assuming the Binomial Theorem. We prove that the theorem is true for n=0, assume it is true for n, and show it is also true for (n+1), thus proving it by induction.

We wish to prove:
[1.1]
Where n is a nonnegative integer for values 0 to n.

## Step1

We wish to prove that the theorem works for n=0.
[1.2]

Which is 1. If there is an objection to 0 over 0 being 1 (or 0 choose 0 is 1), then we can prove the theorem for n=1, 2..n. However, the Binomial Coefficient 0 over 0 is definitely 1 by definition. All the terms after term 0 are zero, because k>n

[1.3]
Similarly, all the terms after term 1 are zero because k>1.

We assume:
[1.1, repeated]

## Step 3

And seek the value of (a+b)n+1 by multiplying 1.1 by (a+b)
[1.2]
We multiply (a+b)n by (a+b):
[1.3]
[1.4]
Adding  these, term by term, we find:
[1.5]

[1.6]

We find that if 1.1 is true then so is 1.2, completing our final induction step.

Ken Ward's Mathematics Pages

# Faster Arithmetic - by Ken Ward

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: