Home
Class 12
MATHS
If m is the AM of two distinct real numb...

If m is the AM of two distinct real numbers l and n `(l,ngt1)` 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

A

`4l^(2)m^(2)n^(2)`

B

`4l^(2)mn`

C

`4lm^(2)n`

D

`4lmn^(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given, `m=(l+n)/2rArr l+n =2n" ...(i)"`
and l, `G_1, G_2, G_3,` n are in GP
`:." "G_1/l =G_2/G_1=G_2/G_2=n/G_3`
`rArr" "G_1G_3 = "In," G_1^2 = lG_2, G_2^2 =G_3 G_1, G_3^2 =nG_21" (ii)"`
Now, `G_1^4 +G_2^4 +G_3^4 =l^2 G_2^2+2G_2^4+n^2G_2^2`
`=G_2^2(l^2 +2G_2^2 +n^2)" "["from eq.(ii)"]`
`=G_3 G_1 (l^2 +2G_2^2 +n^2) " " [from Eq.(ii) ]`
`=G_3 G_1 (l^2 +2G_3 G_1 + n^2)`
= In `(l^2+"2 In"+n^2)" " [from Eq.(ii)]`
= In `(l+n)^2` = In `(2m)^2" "[from Eq. (i)]`
`=4//m^2n`
Promotional Banner

Similar Questions

Explore conceptually related problems

If A is the arithmatic mean and G_(1) and G_(2) be two geometric means between any two numbers then prove that, (G_(1)^(2))/(G_(2)) + (G_(2)^(2))/(G_(1))=2A

If A is the arithmetic mean and G_(1), G_(2) be two geometric mean between any two numbers, then prove that 2A = (G_(1)^(2))/(G_(2)) + (G_(2)^(2))/(G_(1))

Let A_1 , G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n >2, let A_(n-1),G_(n-1) and H_(n-1) has arithmetic, geometric and harmonic means as A_n, G_N, H_N, respectively.

Let A_1 , G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n >2, let A_(n-1),G_(n-1) and H_(n-1) has arithmetic, geometric and harmonic means as A_n, G_N, H_N, respectively.

Let A_1 , G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n >2, let A_(n-1),G_(n-1) and H_(n-1) has arithmetic, geometric and harmonic means as A_n, G_N, H_N, respectively.

Let A_(1),A_(2),A_(3),"......."A_(m) be arithmetic means between -3 and 828 and G_(1),G_(2),G_(3),"......."G_(n) be geometric means between 1 and 2187. Product of geometric means is 3^(35) and sum of arithmetic means is 14025. The value of n is

Let A_(1),A_(2),A_(3),"......."A_(m) be arithmetic means between -3 and 828 and G_(1),G_(2),G_(3),"......."G_(n) be geometric means between 1 and 2187. Product of geometric means is 3^(35) and sum of arithmetic means is 14025. The value of m is

A_1 and A_2 are arithmetic mean between a and b also G_1 and G_2 are geometric mean then (G_(1).G_(2))/(A_(1)+A_2) = ............ .

The A.M. of two positive number is 2. If 1 is added in large number then their G.M. will become 2. The numbers are…….