Question

The mean of first n odd natural numbers is \[\frac{n^2}{81}\],then n =

Options

• 9

• 81

• 27

• 18

Answer 1

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

∴ Mean of first n odd natural numbers

\[= \frac{1 + 3 + 5 + . . . + \left( 2n - 1 \right)}{n}\]

\[ = \frac{\frac{n}{2}\left( 1 + 2n - 1 \right)}{n} \left[ S_n = \frac{n}{2}\left( a + l \right) \right]\]

\[ = \frac{2n}{2}\]

\[ = n\]

Now,
Mean of first n odd natural numbers = \[\frac{n^2}{81}\] (Given)

\[\therefore n = \frac{n^2}{81}\]

\[ \Rightarrow n = 81\]