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(k) \Rightarrow S(k + 1)
• C. S(k)\overset{\not{}}{\Rightarrow}S(k + 1)
• D. S(1) is correct

Answer 1

Answer: (C)

S(k)\overset{\not{}}{\Rightarrow}S(k + 1)

Answer 2

We have $S(k) = 1 + 3 + 5 + ...... + (2k - 1) = 3 + k^{2}$,

$S(1) \Rightarrow 1 = 4$, Which is not true and

$S(2) \Rightarrow 3 = 7$, Which is not true.

Hence induction cannot be applied and $S(k) \Rightarrow S(k + 1)$