Question

Find the arithmetic mean of the first 100 natural numbers?
(a) 50
(b) 52
(c) 51
(d) 50.5

Answer 1

We start solving the problem by recalling the definition of arithmetic mean of given ‘n’ numbers as $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$. We then use this definition for the first 100 natural numbers. We then use the sum of first ‘n’ natural numbers as $\dfrac{n\left( n+1 \right)}{2}$ for the sum in the numerator. We then make the necessary calculations to the required value of mean.

Complete step by step answer: :
According to the problem, we have to find the arithmetic mean of the first 100 natural numbers.
We know that the arithmetic mean of ‘n’ numbers ${{a}_{1}}$, ${{a}_{2}}$, ……, ${{a}_{n}}$ is defined as $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$ ---(1).
We know that the first 100 natural numbers are 1, 2, 3, ……, 100. Let us substitute these numbers in equation (1).
So, we get the arithmetic mean of the first 100 natural numbers = $\dfrac{1+2+3+......+100}{100}$.
We know that the sum of first ‘n’ natural numbers is defined as $\dfrac{n\left( n+1 \right)}{2}$.
$\Rightarrow$ Arithmetic mean of first 100 natural numbers = $\dfrac{\dfrac{100\times \left( 101 \right)}{2}}{100}$.
Arithmetic mean of first 100 natural numbers = $\dfrac{101}{2}$.
Arithmetic mean of first 100 natural numbers = 50.5.
So, we have found the arithmetic mean of the first 100 natural numbers as 50.5.

The correct option for the given problem is (d).

Note:
We should know that the arithmetic mean of ‘n’ numbers is similar to the mathematical average of that ‘n’ numbers. We can also use the fact that the arithmetic mean of first ‘n’ natural numbers is $\dfrac{\left( n+1 \right)}{2}$ to solve this problem. We should not make calculation mistakes while solving this problem. Similarly, we can expect problems to find the harmonic mean and geometric mean for the given numbers.