If m is the A.M. of two distinct real numbers `l` and `n""(""l ,""n"">""1)` and G1, G2 and G3 are three geometric means between `l` and n, then `G1 4+2G2 4+G3 4` equals, (1) `4l^2` mn (2) `4l^m^2` mn (3) `4l m n^2` (4) `4l^2m^2n^2`

Options

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

Answer

B

Explanation

`m=(l+n)/2`
and `l,G_(1),G_(2),G_(3),n` are in G.P.
`therefore r=(n/l)^(1//4)`
`thereforeG_(1)=l(n/l)^(1//4),G_(2)=l(n/l)^(1//2),G_(3)=l(n/l)^(3//4)`
`(G_(1))^(4)+2(G_(2))^(4)+(G_(3))^(4)`
`=l^(4)(n/l)+2l^(4)(n/l)^(2)+l^(4)(n/l)^(3)`
`l^(3)n+2l^(2)n^(2)+ln^(3)`
`=nl(l^(2)+2nl+n^(2))`
`=nl(l+n)^(2)`
`=4m^(2)nl`