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Question

Prove that the product of n geometric means between two positive numbers is the nth power of the single G.M. between them.

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Solution

New common ratio

r=n+1ba

G.M.s are a(ba)1n+1, a(ba)2n+1, a(ba)3n+1, .................................a(ba)nn+1

Their product would be

a(ba)1+2+3+...........nn+1

=a(ba)n(n+1)2(n+1)

Single GM would be a(ba)12 whose nth power clearly equals our product.

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