In the previous article of Arithmetic Progression, we learnt about the N

^{th }term of series.We learnt how to find the twelfth term of the series or which term
will be 30.

In this article, we will learn the sum
of N term of the series. First, we discuss its basic terms

**.**

**Sum of N**^{ }term

**Sum of N**

^{ }termThis is
the sum of the first N term of Arithmetic Progression. It is denoted by S

_{n}. Where n may be any place in the series of A.P.Some other terms & Detailed explanation on Arithmetic Progression we discussed in our previous article. You should read it for
better understanding

**.**

**1.) The formula for the sum of N term of Arithmetic Progression**

**S**_{n }= (n÷2)(2a + (n-1)d)

Where,

Sâ‚™ = Sum of N term of A.P.

n = number of terms or position of term.

a = first term of A.P.

d = common difference.

**2.) How to find the sum of N term in Arithmetic Progression (A.P.)?**

**2.) How to find the sum of N term in Arithmetic Progression (A.P.)?**

Take an
example to find the sum of N terms of an Arithmetic Progression. Let's
discuss it

**-**

**Find the sum of first 22 terms of Arithmetic Progression****2,4,6,8,10.........**

*Given-*

**a = 2, d = 2**(4-2)**& n = 22**,

*Finding*

**Sâ‚™=?**means S_{22 }because we need to find the sum of first 22 terms where n shows a number of terms or position of term. For this, we have a formula**-**

**S**_{n }= (n÷2)(2a + (n-1)d)*Put all the values in above formula :*

**S**_{n }= (n÷2)(2a + (n-1)d)*S*_{22 }= (22÷2)(2×2 + (22-1)2)*S*_{22 }= 11(4 + 21×2)*S*_{22 }= 11( 4 + 42)*S*_{22 }= 11( 46 )*S*_{22 }= 506

So here
the sum of the first 22 terms

**S₂₂**in given A.P. is 506.**3.) How to find the value of n in A.P. for the given sum?**

Now we
take another example for how to find the position of term or value of N if we
have the sum of N term means S

_{n}.Let's discuss it

**-**

*Find the value of n if A.P. is 2,4,6,8,....... & sum of first n term is 1000.*

*Given-*

**S**( We need to find how many terms give sum 1000 in given A.P.). Put all the above values in formula :_{n}= 1000, a = 2, d = 2, n = ?**S**_{n }= (n÷2)(2a + (n-1)d)*1000 = (n÷2)(2×2 + (n-1)2)**1000 = (n÷2)(4+2n-2)**1000 = (n÷2)(2+2n)**1000×2 = n(2+2n)**2000 = 2n + 2n²**2n²*^{ }+ 2n - 2000 = 0

*Solving the above quadratic equation by the quadratic formula-*

*Above solution indicating there is not any value of N which give the sum 1000 in given A.P.*

*If n is any natural number then we consider it. But here we get decimal value so we will not consider.*

From the above examples, we understand how to find the sum of N term & value of N for the given sum in Arithmetic Progression (A.P.).

We may find the value of the common difference in an Arithmetic
Progression.