# The arithmetic mean of the first n natural numbers is

$

A)\dfrac{{\left( {n + 1} \right)}}{2} \\

B)\dfrac{{\left( {n - 1} \right)}}{2} \\

C)\dfrac{n}{2} \\

D){\text{none of the above}} \\

$

Answer

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326.4k+ views

Hint: To proceed with a solution we need the sum of n natural numbers which will be helpful to solve the arithmetic mean of the first n natural numbers where the formula of arithmetic mean of the first n natural numbers includes the sum of n natural numbers.

We know that

First n natural numbers are 1, 2, 3, 4……………………..n

We also know that sum of n natural numbers =$\dfrac{{n\left( {n + 1} \right)}}{2}$

Hence we know that

Arithmetic mean of n natural numbers = $\dfrac{{{\text{Sum of natural numbers}}}}{{{\text{Total natural numbers}}}}$

Arithmetic mean = $\dfrac{{\dfrac{{n\left( {n + 1} \right)}}{2}}}{n}$

Arithmetic mean = $\dfrac{{n + 1}}{2}$

Therefore arithmetic mean is $\dfrac{{n + 1}}{2}$

Option A is the correct answer.

NOTE: In this problem to get the arithmetic mean of n natural numbers we need to know the sum of n natural numbers where the formula of arithmetic mean includes .After getting the values we have substituted in formula and proceeded on calculation part. Here the calculation is a simple cancellation of n terms where we get the answer as $\dfrac{{n + 1}}{2}$.

We know that

First n natural numbers are 1, 2, 3, 4……………………..n

We also know that sum of n natural numbers =$\dfrac{{n\left( {n + 1} \right)}}{2}$

Hence we know that

Arithmetic mean of n natural numbers = $\dfrac{{{\text{Sum of natural numbers}}}}{{{\text{Total natural numbers}}}}$

Arithmetic mean = $\dfrac{{\dfrac{{n\left( {n + 1} \right)}}{2}}}{n}$

Arithmetic mean = $\dfrac{{n + 1}}{2}$

Therefore arithmetic mean is $\dfrac{{n + 1}}{2}$

Option A is the correct answer.

NOTE: In this problem to get the arithmetic mean of n natural numbers we need to know the sum of n natural numbers where the formula of arithmetic mean includes .After getting the values we have substituted in formula and proceeded on calculation part. Here the calculation is a simple cancellation of n terms where we get the answer as $\dfrac{{n + 1}}{2}$.

Last updated date: 27th May 2023

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