**Correct option (a) n**^{2}(n + 1)/2

**Explanation:**

1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + ........ + (2k - 1)^{2} + 2(2k)^{2}

^{= }2k(2k + 1)^{2}/2

Let n = 2k + 1 (odd)

1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + ........ + (2k - 1)^{2} + 2(2k)^{2} + (2k + 1)^{2}

= 2k(2k + 1)^{2} /2 + (2k + 1)^{2}

= (2k + 1)^{2} (2k + 2)/2 = n^{2}(n + 1)/2