Question

Find the sum of first n natural number.

Answer 1

Hint: The sum of n natural number means we have to find the sum of first n positive terms of the series. As we know that, the terms in natural series are in arithmetic form so we can find the sum by using the arithmetic sum formula which is: $ {S_p} = \dfrac{p}{2}\left( {2a + \left( {n - 1} \right)d} \right) $

Complete step-by-step answer:
Here, we have to take the first n natural number.
As the natural numbers start with 1 so the series is: 1, 2, 3, 4… n.
Now looking at the series we get,
The first term of the series is 1.
The common difference between the series is: $ {\rm{second}}\;{\rm{term - first}}\;{\rm{term = 2 - 1 = 1}} $
The total number of terms taken in the series for the sum is n.
Hence the sum of the n terms by using the formula $ {S_p} = \dfrac{p}{2}\left( {2a + \left( {n - 1} \right)d} \right) $
 Where, p is the number of terms of series, a is the first term and d is the common difference of the series.
Now, substituting the values in the equation or formula then we get,
${S_n} = \dfrac{n}{2}\left( {2 \times 1 + \left( {n - 1} \right) \times 1} \right)$
Solving the equation by using the BODMAS method so first we open the brackets given in the equation,
 $
\Rightarrow {S_n} = \dfrac{n}{2}\left( {2 + n - 1} \right)\\
\Rightarrow {S_n} = \dfrac{n}{2}\left( {n + 1} \right)\\
\Rightarrow {S_n} = \dfrac{{n \times \left( {n + 1} \right)}}{2}
 $
Therefore, the sum of n natural number is:
 $ {S_n} = \dfrac{{n\left( {n + 1} \right)}}{2} $

Note: We can place any value in place of n so it will give the accurate sum of natural numbers till that value. It is the standard form of the sum of series having natural numbers with a common difference of 1. It can be directly applied in the problems having natural numbers with 1 as the common difference between two consecutive terms of the series.