Question

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

• A. $S\left(1\right)$ is correct
• B. $S\left(K\right)\Rightarrow S(K+1)$
• C. $S\left(K\right)\ne$ S(K$+$1)
• D. Principle of mathematical induction can be used to prove the formula

Answer 1

The correct option is B $S\left(K\right)\Rightarrow S(K+1)$

$S\left(K\right)=1+3+5+...+(2K-1)=3+K^2$

Put $K=1$ in both sides

$\therefore L.H.S=1$ and $R.H.S=3+1=4$

$\Rightarrow L.H.S\ne R.H.S$

Put $(K+1)$ on both sides in the place of $k$

$L.H.S=1+3+5+...+(2K-1)+(2K+1)$

and $R.H.S=3+{(K+1)}^2=3+K^2+2K+1$

Let $L.H.S=R.H.S$

$\Rightarrow 1+3+5+...+(2K-1)+(2K+1)=3+K^2+2K+1\Rightarrow 1+3+5+...+(2K-1)=3+K^2$

If $S\left(K\right)$ is true, then $S(K+1)$ is also true.

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