Question

The sum of n arithmetic means between a and b is ___________.

Answer 1

Let a and b be two given numbers
Let A₁, A₂, ______ A_n be arithmetic means between a and b.
∴ a, A₁, A₂, _____ A_n , b are in A.P
Here total number of terms is n + 2 first term is a last term is b.
∴ Sum of first n + 2 terms = $\frac{\left(n+2\right)}{2}\left\{a+b\right\}$
$$\begin{gathered} \mathrm{i}.\mathrm{e}a+A_1+A_2+..........+A_n+b=\left(\frac{n+2}{2}\right)\left(a+b\right) \\ \mathrm{i}.\mathrm{e}{A}_1+A_2+........+A_n=\frac{\left(n+2\right)\left(a+b\right)}{2}-a-b \\ =\frac{\left(n+2\right)\left(a+b\right)}{2}-\left(a+b\right) \\ =\left(a+b\right)\left\{\frac{n+2}{2}-1\right\} \\ =\left(a+b\right)\left\{\frac{n+2-2}{2}\right\} \\ =\left(a+b\right)\frac{n}{2} \\ \mathrm{i}.\mathrm{e}{A}_1+A_2+........+A_n=n\left(\frac{a+b}{2}\right) \end{gathered}$$
i.e