Question

If $a,b,c,d,e$ are five positive numbers then

• A. $\left(\frac{a}{b}+\frac{b}{c}\right)(\frac{c}{d}+\frac{d}{e})\ge 4.\sqrt{\frac{a}{e}}$
• B. $\left(\frac{a}{b}+\frac{c}{d}\right)(\frac{b}{c}+\frac{d}{e})\ge 4.\sqrt{\frac{a}{e}}$
• C. $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{e}+\frac{e}{a}\ge 5$
• D. $\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{e}{d}+\frac{a}{e}\ge \frac{1}{5}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\left(\frac{a}{b}+\frac{b}{c}\right)(\frac{c}{d}+\frac{d}{e})\ge 4.\sqrt{\frac{a}{e}}$
B $\left(\frac{a}{b}+\frac{c}{d}\right)(\frac{b}{c}+\frac{d}{e})\ge 4.\sqrt{\frac{a}{e}}$
C $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{e}+\frac{e}{a}\ge 5$

using the result $A.M\ge G.M$
$\frac{a}{b}+\frac{b}{c}\ge 2\sqrt{\frac{a}{c}}\frac{c}{d}+\frac{d}{e}\ge 2\sqrt{\frac{c}{e}}$
$\Rightarrow \left(\frac{a}{b}+\frac{b}{c}\right)\left(\frac{c}{d}+\frac{d}{e}\right)\ge 4\sqrt{\frac{a}{e}}$

$\frac{a}{b}+\frac{c}{d}\ge 2\sqrt{\frac{ac}{bd}}\frac{b}{c}+\frac{d}{e}\ge 2\sqrt{\frac{bd}{ce}}$
$\Rightarrow \left(\frac{a}{b}+\frac{c}{d}\right)\left(\frac{b}{c}+\frac{d}{e}\right)\ge 4\sqrt{\frac{a}{e}}$

Consider $\frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{e}+\frac{e}{a}}{5}\ge 1$

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

Hence, options $A,B,C$.