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 1

$$\begin{gathered} \mathrm{Let}{A}_1,A_2......A_n\mathrm{be}n\mathrm{A}.\mathrm{M}.\mathrm{s}\mathrm{between}\mathrm{two}\mathrm{numbers}a\mathrm{and}b. \\ \mathrm{Then},a,A_1,A_2.......A_n,b\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}.\mathrm{with}\mathrm{common}\mathrm{difference},d=\frac{b-a}{n+1}. \\ \therefore A_1+A_2+......+A_n=\frac{n}{2}\left[A_1+A_n\right] \\ =\frac{n}{2}\left[A_1-d+A_n+d\right] \\ =\frac{n}{2}\left[a+b\right] \\ =n\times \left[\frac{a+b}{2}\right] \\ =\mathrm{A}.\mathrm{M}.\mathrm{between}a\mathrm{and}b,\mathrm{which}\mathrm{is}\mathrm{constant}. \end{gathered}$$