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.

Step2

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]

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