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Geometric progression

If m is the AM of two distinct real numbers l and n(l, n>1) an

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Hard

Solving time: 4 mins

If $m$ is the AM 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

4 l^{2} mn

4 l m^{2} n

4 m n^{2}

4 l^{2} m^{2} n^{2}

Views: 6,032 students

Updated on: Sep 16, 2023

Book:

Arihant JEE Main Chapterwise Solutions Mathematics (2019-2002) (Arihant)

Exam:

JEE Mains 2015

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Mateo Discussed

m

is the AM 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

14 mins ago

Discuss this question LIVE

14 mins ago

Text solutionVerified

Exp. (b)
Given,

m

is the AM of

l

and

n

∴ l + n = 2 m

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} = l r

G_{2} = l r^{2}

G_{3} = l r^{3}

n = l r^{4}

r = (\frac{n}{l})^{\frac{1}{4}}

Now, G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4} = (l r)^{4} + 2 (l r^{2})^{4} + (l r^{3})^{4}

= l^{4} \times r^{4} (1 + 2 r^{4} + r^{8})

= l^{4} \times r^{4} (r^{4} + 1)^{2}

= l^{4} \times \frac{n}{l} (\frac{n + l}{l})^{2}

= l n \times 4 m^{2} = 4 l m^{2} n

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Question Text	If $m$ is the AM 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
Updated On	Sep 16, 2023
Topic	Sequences and Series
Subject	Mathematics
Class	Class 11
Answer Type	Text solution:1 Video solution: 6
Upvotes	761
Avg. Video Duration	3 min

If $m$ is the AM 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

Text solutionVerified

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If $m$ is the AM 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