Question

Let $S\left(K\right)=1+3+\mathrm{5...}+(2K-1)=3+K^2$. 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. $S\left(K\right)\text{/}\Rightarrow S(K+1)$
• D. S(1) is correct

Answer 1

Answer: (B)

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

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

$S\left(1\right)=1=3+1,$ which is not true

$\because$ S(1) is not true.

$\therefore$ P.M. 1 cannot be applied

Let $S\left(k\right)$ is true, i.e.

$1+3+\mathrm{5....}+(2k-1)=3+k^2$

$\Rightarrow 1+3+\mathrm{5....}+(2k-1)+2k+1$

$=3+k^2+2k+I=3+(k+1{)}^2$

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