If m is the A.M. of two distinct real numbers $\ell$ and $\mathrm{n}(\ell, \mathrm{n}>1)$ and $\mathrm{G}_{1}, \mathrm{G}_{2}$ and $\mathrm{G}_{3}$ are three geometric means between $\ell$ and n , then $\mathrm{G}_{1}^{4}+2 \mathrm{G}_{2}^{4}+\mathrm{G}_{3}^{4}$ equals :

(C) $4 \ell^{2} \mathrm{~m}^{2} \mathrm{n}^{2}$

JEE Main · 2015 · 04 Apr 2015 · Offline · Sequences and Series

If m is the A.M. 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} + 2 G_{2}^{4} + G_{3}^{4}

equals :

Answer: (A)

The arithmetic mean gives

ℓ + n = 2 m

. Slotting three geometric means between

ℓ

and

n

makes the five numbers

ℓ, G_{1}, G_{2}, G_{3}, n

a geometric progression with ratio

r = (n / ℓ)^{1/4}

, so

G_{1} = ℓ r

G_{2} = ℓ r^{2}

G_{3} = ℓ r^{3}

. Then

G_{1}^{4} = ℓ^{3} n

G_{2}^{4} = ℓ^{2} n^{2}

and

G_{3}^{4} = ℓ n^{3}

, so

G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4} = ℓ n (ℓ^{2} + 2 ℓ n + n^{2}) = ℓ n (ℓ + n)^{2}

. Replacing

(ℓ + n)^{2}

(2 m)^{2}

turns this into

4 ℓ m^{2} n

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option