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Question

If one geometric mean is G and two arithmetic means are p and q, prove that : G2=(2pq)(2qp)

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Solution

Let the two numbers are a and b,
Geometric mean, G=ab

To insert two arithmetic means between a and b,
Common difference, d=ba3

nth arithmetic mean An=a+nd

p=a+d=a+ba3

q=a+2d=a+2×ba3

Now,
(2pq)(2qp)=(a)(b)

(2pq)(2qp)=(ab)2=G2

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