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Question

Prove that the sum of the n arithmetic means inserted between two quantities is n times the single arithmetic mean between them.

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Solution

a, x1,x2,..xn, b be is an A.P. of n+2 terms where x1,x2,...xn are n A.M.'s between a and b
Now x1+x2+x3+...+xn.
=n2(x1+xn)=n2(T2+Tn+1)
=n2(a+d+bd)=n(a+b2)
=n (sngle A.M. of a and b).

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