# Write the formula for the sum of first n positive integers.

Answer

Verified

365.7k+ views

Hint:- Check whether n numbers form an A.P or not and apply $ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ .

For finding the sum of first n positive integers.

As we know that the first positive integer is 1.

And the second positive integer will be 2.

And this sequence goes on till the last number is n.

So, first n positive numbers will be \[1,2,3,4......n - 1,n\].

As we see that these n numbers written above forms an A.P. With,

$ \Rightarrow $First number, $a = 1$.

$ \Rightarrow $Common difference,${\text{ }}d = 1$.

$ \Rightarrow $And total number of terms $ = n$.

As, we know that the sum of n terms of an A.P is given as ${S_n}$. Where,

$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ (1)

So, putting the values of a, d and n in equation 1. We get,

$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2 + \left( {n - 1} \right)} \right) = {\text{ }}\dfrac{{n\left( {n + 1} \right)}}{2}{\text{ }}$

Hence, sum of first n positive integers will be $\dfrac{{n\left( {n + 1} \right)}}{2}{\text{ }}$

Note:- Whenever we came up with this type of problem where we are asked to find the

Sum or product of some numbers then, first we had to check whether these number have any

relation with each other, because in most of the cases they form A.P or G.P. Then we can easily

find the sum or product of these numbers using A.P or G.P formula.

For finding the sum of first n positive integers.

As we know that the first positive integer is 1.

And the second positive integer will be 2.

And this sequence goes on till the last number is n.

So, first n positive numbers will be \[1,2,3,4......n - 1,n\].

As we see that these n numbers written above forms an A.P. With,

$ \Rightarrow $First number, $a = 1$.

$ \Rightarrow $Common difference,${\text{ }}d = 1$.

$ \Rightarrow $And total number of terms $ = n$.

As, we know that the sum of n terms of an A.P is given as ${S_n}$. Where,

$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ (1)

So, putting the values of a, d and n in equation 1. We get,

$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2 + \left( {n - 1} \right)} \right) = {\text{ }}\dfrac{{n\left( {n + 1} \right)}}{2}{\text{ }}$

Hence, sum of first n positive integers will be $\dfrac{{n\left( {n + 1} \right)}}{2}{\text{ }}$

Note:- Whenever we came up with this type of problem where we are asked to find the

Sum or product of some numbers then, first we had to check whether these number have any

relation with each other, because in most of the cases they form A.P or G.P. Then we can easily

find the sum or product of these numbers using A.P or G.P formula.

Last updated date: 25th Sep 2023

â€¢

Total views: 365.7k

â€¢

Views today: 7.65k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

The poet says Beauty is heard in Can you hear beauty class 6 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE