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Let $S(K)=1+3+5+\ldots+(2 K-1)=3+K^2$. Then which of the following is true?
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Verified Answer
The correct answer is:
$\mathrm{S}(\mathrm{K}) \Rightarrow \mathrm{S}(\mathrm{K}+1)$
$\mathrm{S}(\mathrm{K}) \Rightarrow \mathrm{S}(\mathrm{K}+1)$
$$
\begin{aligned}
& S(k)=1+3+5+\ldots \ldots . .+(2 k-1)=3+k^2 \\
& S(k+1)=1+3+5+\ldots \ldots \ldots+(2 k-1)+(2 k+1) \\
& =\left(3+k^2\right)+2 k+1=k^2+2 k+4\left[\text { from } S(k)=3+k^2\right] \\
& =3+\left(k^2+2 k+1\right)=3+(k+1)^2=S(k+1)
\end{aligned}
$$
Although $S(k)$ in itself is not true but it considered true will always imply towards $S(k+1)$.
\begin{aligned}
& S(k)=1+3+5+\ldots \ldots . .+(2 k-1)=3+k^2 \\
& S(k+1)=1+3+5+\ldots \ldots \ldots+(2 k-1)+(2 k+1) \\
& =\left(3+k^2\right)+2 k+1=k^2+2 k+4\left[\text { from } S(k)=3+k^2\right] \\
& =3+\left(k^2+2 k+1\right)=3+(k+1)^2=S(k+1)
\end{aligned}
$$
Although $S(k)$ in itself is not true but it considered true will always imply towards $S(k+1)$.
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