Question

Let $S\left(K\right)=1+3+5+.....+(2K-1)=3+K^2$. Then which of the following is true?

• A. $S\left(1\right)$ is correct.
• B. The principle of mathematical induction can be used to prove the formula.
• C. $S\left(K\right)$ does not imply $S(K+1)$
• D. $S\left(K\right)\Rightarrow S(K+1)$

Answer 1

The correct option is D

$S\left(K\right)\Rightarrow S(K+1)$

$S\left(K\right)=1+3+5+...+(2K-1)=3+K^2$
Putting $K=1$ in both sides, we get
$L.H.S\ne R.H.S$
Putting $(K+1)$ on both sides in place of $K$, w get
$L.H.S.=1+3+5+...+(2K-1)+(2K+1)$
$R.H.S=3+(K+1{)}^2=3+K^2+2K+1$
Let $L.H.S.=R.H.S.$ Then,
$1+3+5+...+(2K-1)+(2K+1)=3+K^2+2K+1\Rightarrow 1+3+5+...+(2K-1)=3+K^2$
If $S\left(K\right)$ is true, then $S(K+1)$ is also true.
Hence, $S\left(K\right)\Rightarrow S(K+1)$