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One AM, Two GMs Between a, b: Prove p^3+q^3=2Apq | JEE

JEE Maths question with a full step-by-step solution.

Question

If one A.M.

A

and two G.M.s

p

and

q

are inserted between two numbers

a

and

b

, then which is true?

a^3+b^3=2Apq

p^3+q^3=2Apq

correct

a^3+b^3=2Aab

DNone of these

Solution

Step 1: The one A.M. gives

a+b=2A

. The two G.M.s make

a, p, q, b

a G.P., so

p^2=aq

and

q^2=pb.

Step 2: Multiply these by

p

and

q

respectively:

p^3=apq,\qquad q^3=bpq.

Step 3: Add:

p^3+q^3=pq(a+b)=2Apq.

Correct answer: (2)

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