Question

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

• A. Principle of mathematical induction can be used to prove the formula
• B. $S\left(k\right) \Rightarrow S\left(k +1\right)$
• C. $S(k) not S(k +1)$
• 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.
$1+ 3+ 5....+ (2k -1) = 3 + k^2$
$\Rightarrow 1+ 3+ 5....+ \left(2k -1\right) + 2k +1$
$=3+ k^{2} + 2k +1 = 3+ \left(k +1\right)^{2}$
$\therefore S\left(k\right) \Rightarrow S\left(k+1\right)$