Statement - 1: For every natural number $n \geq 2, \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{n}}>\sqrt{n}$. Statement -2 : For every natural number $n \geq 2, \sqrt{n(n+1)}<n+1$.

(C) Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.

(A) Statement -1 is false, Statement -2 is true

(B) Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement -1

(D) Statement - 1 is true, Statement - 2 is false.

Mathematical Induction — JEE Main · 2008 · 27 Apr 2008 · Offline Question with Solution

Statement - 1: For every natural number

n \geq 2, \frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n} > n

.

Statement -2 : For every natural number

n \geq 2, n (n + 1) < n + 1

(A)Statement -1 is false, Statement -2 is true
(B)Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement -1
(C)Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.✓ Correct
(D)Statement - 1 is true, Statement - 2 is false.

Answer: (C)

The first statement, that

\sum_{k = 1}^{n} \frac{1}{k} > n

for

n \geq 2

, is true and can be shown by induction, the inductive step resting on

n (n + 1) > n

. The second statement,

n (n + 1) < n + 1

, is also true since

n (n + 1) < (n + 1)^{2}

. Both hold, but the inequality actually used to drive the induction is

n (n + 1) > n

rather than the upper bound of the second statement, so the second is true yet is not the correct explanation of the first.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option