If m is the A.M of two distict real numbers l and n `(l, n gt 1)`and `G_1,G_2 and G_3` are three geomatric means between l and n, then `(G_1)^4+2(G_2)^4+(G_3)^4` equals

Options

A. `4lmn^(2)`
B. `4l^(2)m^(2)n^(2)`
C. `4l^(2)mn`
D. `4lm^(2)n`

Answer

D

Explanation

We have, `m=(l+n)/(2)`
Let be the common ratio of the G.P. l, `G_(1),G_(2),G_(3),n`. Then `n=lr^(4)rArrr=((n)/(l))^(1//4)`
`:." "G_(!)=lr=l^(3//4)n^(1//4),G_(2)=lr^(2)=l^(2//4)n^(2//4),G_(3)=lr^(3)=l^(1//4)n^(3//4)`.
`rArr" "G_(1).^(4)+2G_(2).^(4)+G_(3).^(4)=l^(3)n+2l^(2)n^(2)+l n^(3)= ln(l+n)^(2)=l n(2m)^(2)=4lm^(2)n`.