Take the terms in pairs

2^2 - 1^2 = (2 + 1) (2 - 1) = 3 * 1 = 3

4^2 - 3^2 = (4 + 3) (4 - 3) = 7 * 1 = 7

6^2 - 5^2 = (6 + 5) (6 - 5) = 11 * 1 = 11

It's an arithmetic series with difference = 4

2n terms = n pairs

n 

1 ... 2 ... 3 .... 4 ... .... n

3 ... 7 ...11 ... 15 .... (4n-1)

Using formula for sum of arithmetic series 

(first + last) * (1/2 the number of terms)

Sum = (3 + (4n-1)) * n / 2

or

(4n + 2) * n / 2 

or

(2n + 1) * n