Question

If the sum of n arithmetic means between 9 and 51 is 270, then the value of n is ___________.

Answer 1

Let a = 9
b = 51
Given sum of n arithmetic means between 9 and 51 is 270.
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 = $\frac{\left(n+2\right)}{2}\left\{a+b\right\}$
$$\begin{gathered} A_1+A_2+...........+A_n=\frac{n\left(a+b\right)}{2} \\ \mathrm{i}.\mathrm{e}270=n\left(\frac{9+51}{2}\right) \\ \mathrm{i}.\mathrm{e}270=n\left(\frac{60}{2}\right) \\ \mathrm{i}.\mathrm{e}\frac{270}{30}=n \\ \mathrm{i}.\mathrm{e}n=9 \end{gathered}$$
Hence, value of n is 9.