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AM,GM,HM Inequality

If A 1 , A...

Question

If $A_{1}, A_{2}$ be two arithmetic means and $G_{1}, G_{2}$ be two geometric means between two numbers $a$ and $b$ , then $\frac{A_{1} + A_{2}}{G_{1} G_{2}}$ is equal to:

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Solution

# $A_{1}$ and $A_{2}$ are arithmetic means
between a and b,
So, $A_{1} + A_{2} = a + b$
[sum of r term from
beginning & (n-r) th term
from end is equal
to a+b]
# $G_{1}$ and $G_{2}$ are geometric means
between a and b
So, $G_{1} . G_{2} = a b$
[Product of $r^{t h}$ term
from beginning and
$(n - r)^{t h}$ term from end
is equal to ab]
Now, $\frac{A_{1} + A_{2}}{G_{1} \times G_{2}} = \frac{a + b}{a b}$

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