JEE-Mains 2015 (8/30)

Algebra Level 4

If $m$ is the arithmetic mean of two distinct real numbers $l$ and $n$ $(l,n>1)$ and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n$ , then $G_1^4+2G_2^4+G_3^4$ equals :

4lm^2n

4lmn^2

4l^2mn

4l^2m^2n^2

1 solution

Ashrit Ramadurgam
Mar 20, 2016

The three geometric means are:- $(G_1)^4 = l^3 n$ $(G_2)^4 = l^2 n^2$ $(G_3)^4 = l n^3$ The arithmetic mean is $m = \frac{l+n}{2}$ i.e $2m = l+n$ Therefore, $(G_1)^4 + 2(G_2)^4 + (G_3)^4$ $=l^3 n + 2 l^2 n^2 + l n^3$ $=ln(l^2 + 2 l n + n^2)$ $=ln(l+n)^2$ $=ln(2m)^2$ $=4 l m^2 n$

JEE-Mains 2015 (8/30)

1 solution

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