JEE MAIN - Mathematics (2008 - No. 23)
STATEMENT - 1 : For every natural number $$n \ge 2,$$
$$${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .$$$
STATEMENT - 2 : For every natural number $$n \ge 2,$$, $$$\sqrt {n\left( {n + 1} \right)} < n + 1.$$$
Statement - 1 is false, Statement - 2 is true
Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for statement - 1
Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1
Statement - 1 is true, Statement - 2 is false
Explanation
Statements $$2$$ 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 $$\sqrt 2 < \sqrt n \Rightarrow {1 \over {\sqrt 2 }} > {1 \over {\sqrt n }}$$
$$\sqrt 3 < \sqrt n \Rightarrow {1 \over {\sqrt 3 }} > {1 \over {\sqrt n }};$$
$$\sqrt n \le \sqrt n \Rightarrow {1 \over {\sqrt n }} \ge {1 \over {\sqrt n }}$$
Also $${1 \over {\sqrt 1 }} > {1 \over {\sqrt n }}$$
$$\therefore$$ 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 $$2$$ is a correct explanation of statement $$-1.$$
$$ \Rightarrow \sqrt n < \sqrt {n + 1} ,n \ge 2$$ which is true
$$ \Rightarrow \sqrt 2 < \sqrt 3 < \sqrt 4 < \sqrt 5 < - - - - - - \sqrt n $$
Now $$\sqrt 2 < \sqrt n \Rightarrow {1 \over {\sqrt 2 }} > {1 \over {\sqrt n }}$$
$$\sqrt 3 < \sqrt n \Rightarrow {1 \over {\sqrt 3 }} > {1 \over {\sqrt n }};$$
$$\sqrt n \le \sqrt n \Rightarrow {1 \over {\sqrt n }} \ge {1 \over {\sqrt n }}$$
Also $${1 \over {\sqrt 1 }} > {1 \over {\sqrt n }}$$
$$\therefore$$ 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 $$2$$ is a correct explanation of statement $$-1.$$
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