Proving Binomial Theorem using Mathematical Induction

The Binomial Theorem is the perfect example to show how different streams in mathematics are connected to one another: its coefficients have combinatorial roots and can be traced to terms in Pascal’s Triangle, and expansion of binomials to different orders of power can describe probability and combination distributions. The combinatorial proof as under requires no need for proving again, but after learning a method called Mathematical Induction from incessant internet browsing on a late Saturday night, I thought, why not give it a shot?

Mathematical Induction is a method of mathematical proof used to prove an expression true for all natural numbers. The steps are as under:

  1. State the proposition P(n) that needs proving.
  2. The Basis: Show P(n) is true, when n=1.
  3. The Inductive Step:
    1. Assume n=k
    2. If P(k) is true, show that P(k+1) is true
  4. If P(k+1) is true, therefore P(n) is true.

Binomial Theorem Proof w. steps 3 - Copy

(Side-note: It’s not everyday you get to use Q.E.D.)