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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. AA and two G.M.s pp and qq are inserted between two numbers aa and bb, then which is true?
Aa3+b3=2Apqa^3+b^3=2Apq
Bp3+q3=2Apqp^3+q^3=2Apqcorrect
Ca3+b3=2Aaba^3+b^3=2Aab
DNone of these
Solution
Step 1: The one A.M. gives a+b=2Aa+b=2A. The two G.M.s make a,p,q,ba, p, q, b a G.P., so p2=aqp^2=aq and q2=pb.q^2=pb. Step 2: Multiply these by pp and qq respectively:
p3=apq,q3=bpq.p^3=apq,\qquad q^3=bpq.
 Step 3: Add:
p3+q3=pq(a+b)=2Apq.p^3+q^3=pq(a+b)=2Apq.
 Correct answer: (2)
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