Question

If M is the arithmetic mean of two distinct real number l and n (I, n > 1) and G₁, G₂ and G₃ are three geometric means between l and n, then `G_1^4 + 2G_2^4 + G_3^4` is equal to ______.

Options

• 4l²Mn

• 4lM²n

• lMn²

• 3l²M²n²

Answer 1

If M is the arithmetic mean of two distinct real number l and n (I, n > 1) and G₁, G₂ and G₃ are three geometric means between l and n, then `G_1^4 + 2G_2^4 + G_3^4` is equal to 4lM²n.

Explanation:

Given, M is the arithmetic mean of l and n.

∴ l + n = 2M  ...(i)

and G₁, G₂, G₃ are geometric means between l and n.

l, G₁, G₂, G₃, n are in GP.

Let r be the common ratio of the GP.

∴ G₁ = lr, G₂ = lr², G₃ = lr³, n = lr⁴

`\implies` r = `(n/l)^(1//4)`

Now, `G_1^4 + 2G_2^4 + G_3^4 + 2(lr^2)^4 + (lr^3)^4`

= l⁴ × r⁴ (1 + 2r⁴ + r⁸)

= l⁴ × r⁴ (r⁴ + 1)²

= `l^4 xx n/l ((n + l)/l)^2`

= ln × 4m²

= 4lM²n