Verified Solution • 2025–26 Session
moderate
QJEEM08Mab434: JEE Main 2008 Mathematics - Class 11 JEEM08Mab434 Solution: STATEMENT - 1 : For every natural number STATEMENT - 2 : For…
STATEMENT - 1 : For every natural number
Multiple Choice Options
- 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
- D Statement - 1 is true, Statement - 2 is false
Step-by-Step Verified Solution
Statements is \sqrt {n\left( {n + 1} \right)} < n + 1,n \ge 2
\Rightarrow \sqrt n < \sqrt {n + 1} ,n \ge 2 which is true
\Rightarrow \sqrt 2 < \sqrt 3 < \sqrt 4 < \sqrt 5 < - - - - - - \sqrt n
Now \sqrt 2 < \sqrt n \Rightarrow {1 \over {\sqrt 2 }} > {1 \over {\sqrt n }}
\sqrt 3 < \sqrt n \Rightarrow {1 \over {\sqrt 3 }} > {1 \over {\sqrt n }};
Also {1 \over {\sqrt 1 }} > {1 \over {\sqrt n }}
Adding all, we get
{1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + ....... + {1 \over n} > {n \over {\sqrt n }} = \sqrt n
Hence both the statements are correct and statement is a correct explanation of statement
\Rightarrow \sqrt n < \sqrt {n + 1} ,n \ge 2 which is true
\Rightarrow \sqrt 2 < \sqrt 3 < \sqrt 4 < \sqrt 5 < - - - - - - \sqrt n
Now \sqrt 2 < \sqrt n \Rightarrow {1 \over {\sqrt 2 }} > {1 \over {\sqrt n }}
\sqrt 3 < \sqrt n \Rightarrow {1 \over {\sqrt 3 }} > {1 \over {\sqrt n }};
Also {1 \over {\sqrt 1 }} > {1 \over {\sqrt n }}
Adding all, we get
{1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + ....... + {1 \over n} > {n \over {\sqrt n }} = \sqrt n
Hence both the statements are correct and statement is a correct explanation of statement