Question

if `m` is the AM of two distinct real number `l and n ` and `G_1,G_2 and G_3` are three geometric means between `l` and `n` , then ` (G_1^4 + G_2^4 +G_3^4 )` equals to
A. `4l^(2)mn`
B. `4lm^(2)n`
C. `lmn^(2)`
D. `l^(2) m^(2) n^(2)`

Answer 1

Correct Answer - B
Given, m is the AM of l and n ltbr. `l + n = 2m` ..(i)
and `G_(1), G_(2), G_(3)` are geometric means between l and n
`l, G_(1), G_(2), G_(3), n` are in GP
Let r be the common ratio of this GP
`:. G_(1) = lr, G_(2) = lr^(2), G_(3) = lr^(3), n - lr^(4) rArr r = ((n)/(l))^((1)/(4))`
Now, `G_(1)^(4) + 2G_(2)^(4) + G_(3)^(4) = (lr)^(4) + 2(lr^(2))^(4) + (lr^(3))^(4)`
`= l^(4) xx r^(4) (1 + 2r^(4) + r^(6)) = l^(4) xx r^(4) (r^(4) + 1)^(2)`
`= l^(4) xx (n)/(l) ((n +l)/(l))^(2) = ln xx 4m^(2) = 4l m^(2) n`