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Let us give a proof of the Binomial Theorem using mathematical induction. We will need to use Pascal's identity in the form:
We aim to prove that
We first note that the result is true for n=1 and n=2: and .
Let be a positive integer with for which the statement is true. So
Now consider the expansion
From Pascal's identity, it follows that
Hence the result is true for . By induction, the result is true for all positive integers .