Question

If one a.m a and two gms p andqbe inserted between two numbers a and b then

Answer 1

Question :-

$\sf {If\ one\ A.M.\ A\ and\ two\ G.M.'s\ p\ and\ q}\\\sf {be\ inserted\ between\ two\ given\ numbers,\ then}\\\\\sf {\frac{\ p^{2}}{q} +\frac{\ q^{2}}{p} =?}$

SoluTion :-

Let the numbers be C & D  and A be the arithmetic mean.

Then,

$\sf {A= \frac{C+D}{2} \Rightarrow 2A=C+D\ .....(1)}$

Let p, q be the GM between C & D

C, p, q, D

$\sf {p^{2}=cq\ and\ q^{2}=PD}$

$\sf {\Rightarrow \frac{p^{2}}{q}=C\ and\ \frac{q^{2}}{p}=D }\\\\\\\\\sf {\Rightarrow \frac{p^{2}}{q}+ \frac{q^{2}}{p}=C+D=2A\ .....(From\ eq.\ 1) }\\\\\\\\\sf {\Rightarrow \frac{p^{2}}{q} + \frac{q^{2}}{p}= 2A}$