We know that mean or average of observations, is the sum of the values of all the observations divided by the total number of observations.
and, we have first n odd natural numbers as
1, 3, …, 2n - 1
Clearly the above series is an AP(Arithmetic progression) with first term, a = 1 and common difference, d = 2
And no of terms is clearly n.
And last term is (2n - 1)
We know, sum of terms of an AP if first and last terms are known is:
\(S_n=\frac{n}{2}(a+a_n)\)
Putting the values in above equation we have sum of series i.e.
\(1+2+3+...+n=\frac{n}{2}(1+2n-1)\)
\(=\frac{n(2n)}{2}=n^{2}...[1]\)
As,
Mean = \(\frac{Sum\,of\,all\,terms}{no\,of\,terms}\)
\(\Rightarrow \) Mean \(=\frac{n^{2}}{n}=n\)
Now, given mean \(=\frac{n^{2}}{81}\)
\(\Rightarrow n= \frac{n^{2}}{81}\)
⇒ n = 81
