Home
Class 12
MATHS
LetA(1),G(1),H(1) denote the arithmetic,...

Let`A_(1),G_(1),H_(1)` denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For `nge2`, 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.
Which of the following statement is correct?

A

`G_(1)gtG_(2)gtG_(3)gt"......."`

B

`G_(1)ltG_(2)ltG_(3)lt"......."`

C

`G_(1)=G_(2)=G_(3)="......."`

D

`G_(1)ltG_(3)ltG_(5)lt"......." `and `G_(2)gtG_(4)gtG_(6)gt"......."`

Text Solution

Verified by Experts

The correct Answer is:
C
`A_(1)=(a+b)/(2),G_(1)=sqrt(ab),H_(1)=(2ab)/(a+b)`
`A_(n)=(A_(n-1)+H_(n-1))/(2),G_(n)=sqrt(A_(n-1)H_(n-1))" and " H_(n)=(2A_(n-1)H_(n-1))/(A_(n-1)+H_(n-1))`
Clearly, `G_(1)=G_(2)=G_(3)="......"=sqrt(ab)`.
Promotional Banner

Similar Questions

Explore conceptually related problems

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.

a,g,h are arithmetic mean, geometric mean and harmonic mean between two positive number x and y respectively. Then identify the correct statement among the following

If m is the A.M of two distinct real number l and n (l,n gt 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) equal :

Let x_(1), x_(2),….., x_(n) be n observations and x be their arithmetic mean. The formula for the standard deviation is given by :

If m is the A.M. of two distinct real numbers l and n ( l , n gt 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 :

Let alpha and beta be two positive real numbers. Suppose A_1, A_2 are two arithmetic means; G_1 ,G_2 are two geometric means and H_1 H_2 are two harmonic means between alpha and beta , then

Write the first three terms of the sequence a_(n)=(-1)^(n-1) 5^(n+1)

For what value of n, (a^(n+1)+b^(n+1))/(a^n+b^n) is the arithmetic mean of a and b?

Let n in N ,n > 25. Let A ,G ,H deonote te arithmetic mean, geometric man, and harmonic mean of 25 and ndot The least value of n for which A ,G ,H in {25 , 26 , n} is a. 49 b. 81 c.169 d. 225