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

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Professor PC Thomas Classes · Answer

(b) : $S(k)=1+3+\ldots .+(2 k-1)=3+k^{2}$ When $k=1$, L.H.S of $S(k) 
eq$ R.H.S of $S(k)$ So $S(1)$ is not true.\ Now $S(k+1)=1+3+5+\ldots .+(2 k-1)+$ $(2 k+1)$ $$\begin{equation*}=3+(k+1)^{2} 	ag{ii} \end{equation*}$$ Let $S(k)$ is true $	herefore 1+3+5+\ldots .+(2 k-1)$ $=k^{2}+3 \Rightarrow 1+3+5+\ldots+(2 k-1)+(2 k+1)$\ $=3+k^{2}+2 k+1=(k+1)^{2}+3$\ $\Rightarrow S(k+1)$ is true $	herefore S(k) \Rightarrow S(k+1)$\