Let S(K):1+3+5+…+(2K−1)=3+k2Then which of the following is true?

$S (1) : 1 = 3 + 1^{2}$ which is not true. Suppose

$S (K)$ is true, then

$1 + 3 + 5 + \dots + (2 K - 1) = 3 + K^{2}$

Adding $(2 K + 1)$ to both the sides, we get

$\begin{array}{l} 1 + 3 + 5 + \dots + (2 K - 1) + (2 K + 1) = 3 + K^{2} + 2 K + 1 \\ = 3 + (K + 1)^{2} \end{array}$

which is $S (K + 1)$

Thus, $S (K) f i S (K + 1) .$

Let $S (K) : 1 + 3 + 5 + \dots + (2 K - 1) = 3 +$ $k^{2}$
Then which of the following is true?