Question

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

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

Answer 1

Answer: (B)

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

Putting $k=1$, we get L.H.S=1 and R.H.S=4. Hence $A$ and $D$ are incorrect.

Now, $S(k+1)=1+3+5+$...$+(2k-1)+(2(k+1)-1)$

$\Rightarrow S(k+1)=S\left(k\right)+(2k+1)$ [Since $S\left(k\right)=1+3+5+$...$+(2k-1)$]

$\Rightarrow S(k+1)=3+k^2+(2k+1)$

$\Rightarrow S(k+1)=3+{(k+1)}^2$

$B$ is correct option.