Question

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

Answer 1

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

Comments

why have u take n=2 , n should be 4 a , A1 , A2, b

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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.