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

(D) $S(K) \Rightarrow S(K+1)$

(B) Principle of mathematical induction can be used to prove the formula

(C) $\mathrm{S}(\mathrm{K}) \nRightarrow \mathrm{S}(\mathrm{K}+1)$

Mathematical Induction — JEE Main · 2004 · 24 Apr 2004 · Offline Question with Solution

Let

S (K) = 1 + 3 + 5 + \dots + (2 K - 1) = 3 + K^{2}

. Then which of the following is true?

Answer: (D)

The genuine sum of the first

K

odd numbers is

K^{2}

, so the proposed formula

S (K) = 3 + K^{2}

is off by

3

. Checking the base case, the left side at

K = 1

1

while the right side is

4

, so the statement is false at

K = 1

and induction cannot establish it. The inductive step, however, is self-consistent: assuming

S (K) = 3 + K^{2}

and adding the next odd number

2 K + 1

gives

3 + K^{2} + 2 K + 1 = 3 + (K + 1)^{2}

, which is exactly

S (K + 1)

. So the implication

S (K) \Rightarrow S (K + 1)

does hold.

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option