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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.



Answer -

Let a and b be thefirst and last terms

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

We know, Mean =(a+b)/2

Mean of A1 andAn = (A1 + An)/2

A1 =a+d

An = a– d

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

= (a+b)/2

AM between A2 andAn-1 = (a+2d+b-2d)/2

= (a+b)/2

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

Hence, AM = (a + b)/2

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