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Question
If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.
Solution
\[\text { Let } A_1 , A_2 . . . . . . A_n \text { be n A . M . s between two numbers a and b } . \]
\[\text { Then, } a, A_1 , A_2 . . . . . . . A_n , \text { b are in A . P . with common difference, d } = \frac{b - a}{n + 1} . \]
\[ \therefore A_1 + A_2 + . . . . . . + A_n = \frac{n}{2}\left[ A_1 + A_n \right]\]
\[ = \frac{n}{2}\left[ A_1 - d + A_n + d \right]\]
\[ = \frac{n}{2}\left[ a + b \right]\]
\[ = n \times \left[ \frac{a + b}{2} \right]\]
\[ =\text { A . M . between a and b, which is constant } .\]
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