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+6 votes
in Mathematics by (7.5k points)

If A1, A2 are two AMs between two numbers a and b, then express (2A1-A2)(2A2-A1) in terms of a and b.

1 Answer

+5 votes
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Best answer

Given :

A1 and A2 are two AM between a and b.

Let d be the common difference of this A.P. Clearly, it contains (n+2) terms = 2+2 = 4 terms
=> b = (n+2)th term = 4th term

Now, Putting the value of n = 2 in A1 and A2 we get,

A1 = (2a+b) / 3 -------------(1)

A2  = (a+2b) / 3 -------------(2)

Fro question :

=> (2A1 – A2)*(2A2-A1) = [2(2a+b)/3-(a+2b)/3] * [2/3(a+2b)-(2a+b)/3]

=> (2A1 – A2)*(2A2-A1) = ab

by (10 points)
why have u take n=2 , n should be 4 a , A1 , A2, b
by (22.5k points)
Since, here , we only include two arithmetic means A1 & A2. Therefore, we have to take n= 2. In this formula, n is the number of inserted arithmetic means.

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