Question

The mean of first n odd natural numbers is $\frac{n^2}{81}$, then n =
(a) 9
(b) 81
(c) 27
(d) 18

Answer 1

The first n odd natural numbers are 1, 3, 5, ... , (2n − 1).

∴ Mean of first n odd natural numbers

$$\begin{gathered} =\frac{1+3+5+...+\left(2n-1\right)}{n} \\ =\frac{{\displaystyle \frac{n}{2}}\left(1+2n-1\right)}{n}\left[S_n=\frac{n}{2}\left(a+l\right)\right] \\ =\frac{2n}{2} \\ =n \end{gathered}$$

Now,
Mean of first n odd natural numbers = $\frac{n^2}{81}$ (Given)
$$\begin{gathered} \therefore n=\frac{n^2}{81} \\ \Rightarrow n=81 \end{gathered}$$

Hence, the correct option is (b).