Find the sum of(upto n terms): 1² + 3² + 5² + 7² + 9²...

We'll notice that we'll have to calculate the sum of the
squares of the odd natural terms.

Let 2k + 1 ne the general
term of the sum. If we'll replace k by values from 0 to n, we'll get each odd natural
number.

We'll raise to square and we'll expand the
binomial:

(2k+1)^2 = 4k^2 + 4k +
1

We'll take the
summation:

1^2 + 2^2 +...n^2 =
Sum (2k+1)^2 = Sum 4k^2 + Sum 4k + Sum 1

Sum
(2k+1)^2 = 4*n*(n+1)*(2n+1)/6 + 4*n(n+1)/2 + n

Sum (2k+1)^2
= [4*n*(n+1)*(2n+1) + 12n*(n+1) + 6n]/6

We'll remove the
brackets:

Sum (2k+1)^2 = [4n(n^2 + 3n + 1) + 12n^2 + 12n +
6n]/6

Sum (2k+1)^2 = (4n^3 + 12n^2 + 4n + 12n^2 + 12n +
6n)/6

We'll combine like
terms:

Sum (2k+1)^2 = (4n^3 + 24n^2 +
22n)/6

Sum (2k+1)^2 = 2n(2n^2 + 12n +
11)/6

Sum (2k+1)^2 = n(2n^2 + 12n +
11)/3

The requested sum of the squares of the
odd natural numbers is: 1^2 + 2^2 + ...n^2 = Sum (2k+1)^2 = n(2n^2 + 12n +
11)/3