Question

Find the average of the first 97 natural numbers.
(a) 47
(b) 37
(c) 48
(d) 49
(e) 49.5

Answer 1

Hint: In this question, we have to find the average of the first 97 natural numbers. Natural numbers are those which start from 1. For calculating the average, we will use the formula for n natural numbers:
$\Rightarrow \dfrac{\dfrac{n\times \left( n+1 \right)}{2}}{n}$
In question, it is given the value of n as 97.

Complete step by step solution:
Let’s discuss the question now.
 Numbers are those entities which are used to measure, to count the things, to label the objects etc. They are expressed using digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. There are again two types of numbers: natural and whole. Natural numbers are those numbers which start from 1. And whole numbers are those numbers which start from 0. Whole numbers contain natural numbers as well. Numbers can be expressed as positive and negative too. Numbers which are positive are expressed without ‘-‘ sign. And numbers which are negative are expressed with the ‘-‘ sign before that number.
Now, come to the question part.
For finding the average of n natural numbers we have a formula:
$\Rightarrow \dfrac{\dfrac{n\times \left( n+1 \right)}{2}}{n}$
We will use the value of n = 97 as per question.
$\Rightarrow \dfrac{\dfrac{97\times \left( 97+1 \right)}{2}}{97}$
Now, form the single fraction by reciprocating the denominator:
$\Rightarrow \dfrac{1}{97}\times \dfrac{97\times 98}{2}$
Reduce into the simplest form, we will get:
$\Rightarrow \dfrac{98}{2}=49$

So, the correct answer is “Option d”.

Note: There is an alternative method of calculating the average of the first 97 natural numbers. As we know that these numbers from 1 to 97 are in AP series. If we see the series it will be like:
1, 2, 3, 4, 5, 6, 7, 8, 9,………………………………………90, 91, 92, 93, 94, 95, 96, 97
For calculating the average, we will sum up the corresponding terms, like the corresponding term for 97 is 1, for 96 it is 2 and so on.
So the formula for average is: $\dfrac{sum\text{ of corresponding term}}{2}$
Let’s take first pair, 97 and 1:
Average = $\dfrac{97+1}{2}$ = $\dfrac{98}{2}=49$
Let’s check again with another pair, 96 and 2:
Average = $\dfrac{96+2}{2}$ = $\dfrac{98}{2}=49$
So, we are getting the same average every time. That means the answer is correct.