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Let x be the arithmetic mean and y, z be the two geometric mea

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Let $x$ be the arithmetic mean and $y, z$ be the two geometric means between any two positive numbers. Then $\frac{y ^{3} + z ^{3}}{x yz}$ = __________.

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Let

x

be the arithmetic mean and

y, z

be the two geometric means between any two positive numbers. Then

\frac{y ^{3} + z ^{3}}{x yz}

= __________.

14 mins ago

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14 mins ago

Text solutionVerified

Given that

x

is the A.M. between

a

and

b

and

y

, and

z

are the G.M.

s

between

a

and

b

where

a

and

b

are positive. Then

a, x, b

are in A.P. So,

x = \frac{a + b}{2}

a, y, z, b

are in G.P. So,

y = a r

and

z = a r^{2}

, where

r = 3 \frac{b}{c}

. Also,

yz = ab

, Now,

\frac{y ^{3} + z ^{3}}{x yz} = \frac{a ^{3} r ^{3} + a ^{3} r ^{6}}{ab ( \frac{a + b}{2} )}

= \frac{a ^{3} \times \frac{b}{a} + a ^{3} \times \frac{b ^{2}}{a ^{2}}}{ab ( \frac{a + b}{2} )}

= \frac{2 ( a ^{2} b + a b ^{2} )}{a ^{2} b + a b ^{2}}

\frac{y ^{3} + z ^{3}}{x yz} = 2

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Question Text	Let $x$ be the arithmetic mean and $y, z$ be the two geometric means between any two positive numbers. Then $\frac{y ^{3} + z ^{3}}{x yz}$ = __________.
Answer Type	Text solution:1
Upvotes	150

Let x be the arithmetic mean and y,z be the two geometric means between any two positive numbers. Then xyzy3+z3​ = __________.

Text solutionVerified

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Let $x$ be the arithmetic mean and $y, z$ be the two geometric means between any two positive numbers. Then $\frac{y ^{3} + z ^{3}}{x yz}$ = __________.