Induction

Mathematical Induction Example 1 --- Sum of Evens

Problem: For any natural number n , 2 + 4 + ... + 2n = n( n + 1 ).

Proof:
Basis Step: If n = 0, then LHS = 0, and RHS = 0 * (0 + 1) = 0 .
Hence LHS = RHS.
Induction: Assume that for an arbitrary natural number n, 0 + 2 + ... + 2n = n( n + 1 ) . -------- Induction Hypothesis
To prove this for n+1,   first try to express LHS for n+1   in terms of LHS for n,   and somehow use the induction hypothesis.
Here let us try
      LHS for n + 1 = 0 + 2 + ... + 2n + 2(n + 1) = (0 + 2 + ... + 2n) + 2(n + 1) .
Using the induction hypothesis, the last expression can be rewritten as
      n( n + 1 ) + 2(n + 1) .
Factoring (n + 1) out, we get
      (n + 1)(n + 2) ,
which is equal to the RHS for n+1.

Thus LHS = RHS for n+1.

End of Proof.