# 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. - Mathematics

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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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.6 | Q 7 | Page 46