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Question
The mean of first n odd natural numbers is \[\frac{n^2}{81}\],then n =
Options
9
81
27
18
Solution
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)
\[\therefore n = \frac{n^2}{81}\]
\[ \Rightarrow n = 81\]
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