Question

If the mean of first n natural numbers is 20, then n = __________.

Answer 1

The first n natural numbers are 1, 2, 3, ..., n.

This forms an AP with a = 1 and d = 1.

Sum of first n natural numbers = 1 + 2 + 3 + ... + n = $\frac{n}{2}\left(1+n\right)=\frac{n\left(n+1\right)}{2}\left[S_n=\frac{n}{2}\left(a+l\right)\right]$

∴ Mean of first n natural numbers = $\frac{\mathrm{Sum}\mathrm{of}\mathrm{first}n\mathrm{natural}\mathrm{numbers}}{n}=\frac{{\displaystyle \frac{n\left(n+1\right)}{2}}}{n}=\frac{n+1}{2}$

Now,

$$\begin{gathered} \frac{n+1}{2}=20\left(\mathrm{Given}\right) \\ \Rightarrow n+1=40 \\ \Rightarrow n=40-1=39 \end{gathered}$$

If the mean of first n natural numbers is 20, then n = _____39_____.