Question

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

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

Answer 1

Answer: (B)

$S(k)=1+3+5+...+(2k-1)=3+k^2$
$S(1):1=3+1$, which is not true
$\because S(1)$ is not true.
$\therefore P.M.I$ cannot be applied
Let $S(k)$ is true, i.e.
$1+3+\mathrm{5....}+(2k-1)=3+k^2$
$\Rightarrow 1+3+\mathrm{5....}+\left(2k-1\right)+2k+1$
$=3+k^2+2k+1=3+{\left(k+1\right)}^2$
$\therefore S\left(k\right)\Rightarrow S\left(k+1\right)$