Advertisement Remove all ads

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.

Advertisement Remove all ads

Solution

\[\text { Let } A_1 , A_2 . . . . . . A_n \text { be n A . M . s between two numbers a and b } . \]

\[\text { Then, } a, A_1 , A_2 . . . . . . . A_n , \text { b are in A . P . with common difference, d } = \frac{b - a}{n + 1} . \]

\[ \therefore A_1 + A_2 + . . . . . . + A_n = \frac{n}{2}\left[ A_1 + A_n \right]\]

\[ = \frac{n}{2}\left[ A_1 - d + A_n + d \right]\]

\[ = \frac{n}{2}\left[ a + b \right]\]

\[ = n \times \left[ \frac{a + b}{2} \right]\]

\[ =\text {  A . M . between a and b, which is constant } .\]

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.6 | Q 7 | Page 46
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×