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Home » 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.
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.
Arithmetic Progressions | CBSE | Class 11 | Exercise 19.6 | Maths | RD Sharma
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.

Answer:

Let a and b be the first and last terms

The series be a, A1, A2, A3, …….., An, b

Mean = (a+b)/2

Mean of A1 and An = (A1 + An)/2

A1 = a+d

An = a – d

AM = (a+d+b-d)/2

AM = (a+b)/2

AM between A2 and An-1 = (a+2d+b-2d)/2

=> (a+b)/2

(a + b)/2 is constant for all such numbers

Hence, AM = (a + b)/2

i

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