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if one arithmetic mean `a` and two geometric mean P and Q be inserted between any two given number then show that P^3 + Q^3 = 2apq

if one arithmetic mean `a` and two geometric mean P and Q be inserted between any two given number then show that P^3 + Q^3 = 2apq

Grade:11

2 Answers

Saurabh Koranglekar
askIITians Faculty 10335 Points
4 years ago
576-527_1.PNG
Vikas TU
14149 Points
4 years ago
Dear student 
Let A is A.M of two positive numbers b and c. So,A=(b+c)/2=>2A=b+c
Let P and Q are between two numbers b and c.
So,b,P,Q,c and r be the common ratio
P=br,Q=br^2,c=br^3. So,r=(c/b)^(1/3)
So,P^3=(br)^3=b^3*(c/b)=b^2*c
And,Q=b^3*(c^2/b^2)=b*(c^2).
(P^3+Q^3)/PQ={bc(b+c)}/bc=b+c=2A(proved)
Good Luck 

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