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  • Let x be the arithmetic mean and y, z be the two geometric means between

Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers, then the value of \frac{y^{3}+z^{3}}{xyz} is :

 

Option: 1

2


Option: 2

3


Option: 3

0.5


Option: 4

1.5


Answers (1)

best_answer

Let two positive numbers a and b

So x = (a+b)/2 and a, y, z, b are in G.P.

If r is the common ratio of this G.P., then b=ar^{3}   

r = \left ( \frac{b}{a} \right )^{\frac{1}{3}}

So,  

\frac{y^{3}+z^{3}}{xyz} = \frac{a^{3}r^{3}+a^{3}r^{6}}{x\left ( ar \right )\left ( ar^{2} \right )}=\frac{a\left ( 1+r^{3} \right )}{x}=\frac{a+b}{\frac{a+b}{2}} = 2

Posted by

Ritika Harsh

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