Question

Find the average of all odd numbers up to $100$.

Answer 1

Step 1: Find the number of odd numbers from $1$ to $100$.

Sum of the odd numbers $=1+3+5+7+9+11+13+....+99$.

The given series is in AP with:

$$\begin{gathered} a=1 \\ l=99 \\ d=2 \end{gathered}$$

So, $n^{\mathrm{th}}$ term of the A.P. is $T_n=a+\left(n-1\right)d$.

$$\begin{gathered} 99=1+\left(n-1\right)2 \\ 99=2n-1 \\ n=50 \end{gathered}$$

Step 2: Find the sum of the series and calculate the average

The Sum of the series in AP is calculated as given below:

$\begin{array}{rcl}{S}_n& =& \frac{n}{2}\left(a+l\right)\\ & =& \frac{50}{2}\left(1+99\right)\\ & =& 2500\end{array}$

Therefore,

$\begin{array}{rcl}Averageofoddnumbers& =& \frac{Sumofallobservations}{Totalobservations}\\ & =& \frac{2500}{50}\\ & =& 50\end{array}$

Hence, the average of odd numbers up to $100$ is $50$.