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Prove that the sum of the n arithmetic means inserted between

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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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Text solutionVerified

x_{1}, x_{2}, .. x_{n}

, b be is an A.P. of

n + 2

terms where

x_{1}, x_{2}, ... x_{n}

are n A.M.'s between a and b
Now

x_{1} + x_{2} + x_{3} + ... + x_{n}

= \frac{n}{2} (x_{1} + x_{n}) = \frac{n}{2} (T_{2} + T_{n + 1})

= \frac{n}{2} (a + d + b - d) = n (\frac{a + b}{2})

=

n (sngle A.M. of a and b).

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Question Text	Prove that the sum of the n arithmetic means inserted between two quantities is n times the single arithmetic mean between them.
Answer Type	Text solution:1
Upvotes	150