If n A M s are inserted between two numbers,prove that the sum of the means equidistant from the beginning - Maths - Sequences and Series

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

Aakash Sharma · Accepted Answer

Let A1, A2 An be n A M between two numbers a and b Then a, A1, A2 An, b is in A P with Common difference (d) = $\frac{b - a}{n + 1}$ So, $A_{1} + A_{2} + A_{n} = \frac{n}{2} [A_{1} + A_{n}] = \frac{n}{2} [A_{1} - d + A_{n} + d] = \frac{n}{2} [a + b] = n \times (\frac{a + b}{2})$ = A M between a and b Which is a 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.