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

Question

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

(\frac{a}{b} + \frac{b}{c}) (\frac{c}{d} + \frac{d}{e}) \geq 4. \frac{a}{e}

(\frac{a}{b} + \frac{c}{d}) (\frac{b}{c} + \frac{d}{e}) \geq 4. \frac{a}{e}

\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} + \frac{e}{a} \geq 5

\frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{e}{d} + \frac{a}{e} \geq \frac{1}{5}

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a, b, c, d, e

are five positive numbers then

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Text solutionVerified

using the result

A . M \geq G . M

\frac{a}{b} + \frac{b}{c} \geq 2 \frac{a}{c} \frac{c}{d} + \frac{d}{e} \geq 2 \frac{c}{e}

\Rightarrow (\frac{a}{b} + \frac{b}{c}) (\frac{c}{d} + \frac{d}{e}) \geq 4 \frac{a}{e}

\frac{a}{b} + \frac{c}{d} \geq 2 \frac{a c}{b d} \frac{b}{c} + \frac{d}{e} \geq 2 \frac{b d}{ce}

\Rightarrow (\frac{a}{b} + \frac{c}{d}) (\frac{b}{c} + \frac{d}{e}) \geq 4 \frac{a}{e}

Consider

\frac{\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} + \frac{e}{a}}{5} \geq 1

\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} + \frac{e}{a} \geq 5

Hence, options

A, B, C

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Question Text	If $a, b, c, d, e$ are five positive numbers then
Answer Type	Text solution:1
Upvotes	150