Base step: When n = 1, you have

LHS = 1

RHS = 1^2 = 1 = LHS.

Base step is proven.

Inductive step: Assume P(k) is true, i.e. that 1 + 3 + 5 + ... + (2k-1) = k^2. Using this you need to show that P(k+1) is true.

For P(k + 1) you are trying to show 1 + 3 + 5 + ... + (2k-1) + (2k+1) = (k + 1)^2.

LHS = 1 + 3 + 5 + ... + (2k - 1) + (2k + 1)

= k^2 + 2k + 1 since 1 + 3 + 5 + ... + (2k - 1) = k^2

= k^2 + k + k + 1

= k(k + 1) + 1(k + 1)

= (k + 1)(k + 1)

= (k + 1)^2

= RHS.

So the Inductive step is proven.