Question

Find the average of the first $20$ natural numbers.

Answer 1

Hint: Average can be defined as the mean of the numbers. In other words, mean is expressed as the sum of all the numbers upon the total number of numbers. First of all we will note down the first twenty natural numbers and then will find an average of the numbers by using the formula.

Complete step by step answer:
First $20$ natural numbers are given by -
$1,2,3,.....20$
An average of the data set can also be defined as the mean of all the natural numbers.
The average of the numbers are from the sample is denoted by $\overline x $ and mathematically it is represented as –
$\overline x = \dfrac{{\sum\limits_1^n {{x_i}} }}{n}$
Place the values in the above expression –
$\overline x = \dfrac{{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20}}{{20}}$
Find the sum of the terms in the above expression –
$\overline x = \dfrac{{210}}{{20}}$
Simplify the above expression finding the division of the terms, common factors from the numerator and the denominator cancels each other.
$\overline x = 11.5$
Therefore, the average of the first 20 natural numbers is $11.5$.

Note:
Alternative method: First $20$ natural numbers are given by -
$1,2,3,.....20$
Since the difference between two consecutive natural numbers is constant, the above series can be said as the series of the Arithmetic progression.
Sum of first twenty terms in the arithmetic progression is given by –
${S_n} = \dfrac{n}{2}[a + l]$
Place the values in the above expression –
${S_{20}} = \dfrac{{20}}{2}[1 + 20]$
Simplify the above expression –
${S_{20}} = \dfrac{{20}}{2}[21]$
Find factors of the terms in the numerator –
${S_{20}} = \dfrac{{2 \times 10}}{2}[21]$
Common factors from the numerator and the denominator cancel each other and therefore remove from the numerator and the denominator of the above expression.
${S_{20}} = 10[21]$
Multiply the above expression on the right hand side-
${S_{20}} = 210$
Now, average is given by the sum of the numbers upon the number of numbers.
Average $ = \dfrac{{210}}{{20}}$
Average $ = 11.5$