Question

Assertion :Let a, b, c, d ,e are five positive number.

$(\frac{a}{b}+\frac{b}{c})(\frac{c}{d}+\frac{d}{e})\ge 4\sqrt{\frac{a}{e}}$

Reason: $\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{e}{d}+\frac{a}{e}\ge 5$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Statement-1

As $A.M.\ge G.M.$

$\frac{a}{b}+\frac{b}{c}\ge \sqrt[2]{2\frac{a}{b}\times \frac{b}{c}}$

Or, $\frac{a}{b}+\frac{b}{c}\ge \sqrt[2]{2\frac{a}{c}}$....(i)

Also $\frac{c}{d}+\frac{d}{e}\ge \sqrt[2]{2\frac{c}{e}}$ ....(ii)
Inequality (i) $\times$ (ii)

$(\frac{a}{b}+\frac{b}{c})(\frac{c}{d}+\frac{d}{e})\ge \sqrt[4]{4\frac{a}{c}\times \frac{c}{e}}$

Or $(\frac{a}{b}+\frac{b}{c})(\frac{c}{d}+\frac{d}{e})\ge \sqrt[4]{4\frac{a}{e}}$

Statement (1) is correct.

Statement (2)
As $A.M.\ge G.M.$

$\therefore \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{e}{d}+\frac{a}{e}\ge 5\sqrt[5]{5\frac{b}{a}\times \frac{c}{b}\times \frac{d}{c}\times \frac{e}{d}\times \frac{a}{e}}$

Or, $\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{e}{d}+\frac{a}{e}\ge 5$
So, statement (2) is correct and is correct explanation for statement (1).