Solve the following

Question:

If $A_{1}, A_{2}$ are the two arithmetic means between two numbers a and $b$ and $G_{1}, G_{1}$ are two geometric means between same two numbers, then $\frac{A_{1}+A_{2}}{G_{1} G_{2}}=$

Solution:

If A1A2 are two arithmetic means between two numbers a and b and G1G2 are geometric means between a and b. Figure

i. e $A_{1}=a+\frac{1}{3}(b-a)=\frac{3 a+b-a}{3}=\frac{2 a+b}{3}$

and $A_{2}=a+\frac{2}{3}(b-a)=\frac{3 a+2 b-2 a}{3}=\frac{a+2 b}{3}$

Also Gand G2 are the geometric means between a and b.

So, $G=a\left(\frac{b}{a}\right)^{\frac{1}{3}}=a^{\frac{2}{3}} b^{\frac{1}{3}}$ and $G_{2}=a\left(\frac{b}{a}\right)^{\frac{2}{3}}=a^{\frac{1}{3}} b^{\frac{2}{3}}$

$\frac{A_{1}+A_{2}}{G_{1} G_{2}}=\frac{\frac{2 a+b}{3}+\frac{2 b+a}{3}}{a b}$

$=\frac{2 a+b+2 b+a}{3 a b}$

$=\frac{3(a+b)}{3 a b}$

$\frac{A_{1}+A_{2}}{G_{1} G_{2}}=\frac{a+b}{a b}$

 

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