use mathematical induction to show that the summation from i=n to 2n-1 (2i+1) = 3n^2 whenever n is a nonnegative integer
The upper limit of the sum is also 2k+1. Think about how this second sum compares to the first.
I'll let you figure out why P(k+1) can be written like that. What extra terms does it contain that P(k) does not?