Given $S(K)$ $= 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}$ When k = 1, S(1): 1 = 3 + 1, L.H.S of S(k) $\ne$ R.H.S of S(k) So S(1) is not true. As S(1) is not true so principle of mathematical induction can not be used. S(K+1) = 1 + 3 + 5... + (2K - 1) + (2K + 1) = 3 + (k + 1) 2 Now let S(k) is true $\therefore$ 1 + 3 + 5 +........(2k - 1) = 3 + k 2 $\Rightarrow$ 1 + 3 + 5 +........(2k - 1) + (2k + 1) = 3 + k 2 + 2k +1 = 3 + (k + 1) 2 $\Rightarrow$ S(k + 1) is true. $\therefore$ S(k) $\Rightarrow$ S(k + 1)