To prove:
The sum of the first n odd numbers is equal to n².

Proof:
When n = 1,
1 = 1² — true.

Assume it is true for n = k,
that is,
1 + 3 + 5 + … + (2k – 1) = k²

Now, for n = k + 1,
the left side becomes:
k² + (2k + 1) = (k + 1)²

So, the result holds for n = k + 1 if it holds for n = k.
Since it is true for n = 1, by mathematical induction,
1 + 3 + 5 + … + (2n – 1) = n²
for all natural numbers n.

Hence proved.