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What is the formula to find the sum of n terms in AP?

  • Last Updated : 11 Aug, 2021

A progression is basically a list of terms ( usually numbers) that follow a particular logical and predictable pattern. There is a certain relation between the two terms in each type of Progression. The predictable nature of Progression helps in forming a generalized formula for that Progression, Formulae include finding the nth term of the series, finding the sum of the series, etc. There are three main types of progressions known,

Types of Progression

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In Mathematics, the progression of numbers can be classified into three specific types mainly:



  1. Arithmetic Progression
  2. Geometric Progression
  3. Harmonic Progression

Let’s learn in detail about the arithmetic progression,

Arithmetic Progression 

Arithmetic Progression is basically a sequence of numbers which exist in such a way that the difference between any two consecutive numbers is a constant value or quantity, that difference is denoted as “d”. The first term in A.P. is denoted as “a” and the last term (for finite series) as “n”. For instance, consider the sequence of even natural numbers 2, 4, 6, 8, 10,…….If we consider the difference between any two numbers (8- 6) is 2. Some of the other few examples of Arithmetic Progression is Sequence of odd natural numbers, Sequence of natural numbers.

A generalized representation of Arithmetic Progression

The first term is represented as “a” and the common difference is represented as “d”, therefore, the next term should is a+d, and the next term to that should be a+d+d, based on this, a generalized way of representing the A.P. can be formed. The Arithmetic Progression can be expressed as,

a, a+d, a+2d, a+3d, a+4d, ………. a+ (n-1)d

In the above expression, “a” represents the first term  of the progression, “d” represents the common difference

The last term “anof the progression is represented as,

an=a+(n-1)d 



What is the formula for the sum of n terms of an A.P?

The Sum of any progression is basically the summation of all its terms, there is a generalized formula formed for the n terms of an A.P. If the first term is denoted as “a”, the common difference is denoted as “d”, the number of terms present is denoted as “n”, then the formula is given as,

S_n= \frac{n}{2}[2a+ (n-1)d]

Or   

The Sum of n terms of an Arithmetic Progression can also be given by Sn ,

Sn= N*[First term+ Last term]/2

Proof for the sum of n terms in an AP

Lets consider the Generalized representation of Arithmetic Progression, the sum of all the terms in the above sequence is given as,

a, a+d, a+2d, a+3d, a+4d, ………. a+ (n-1)d

Sn= (a+ a+ d+ a+ 2d+ a+ 3d+ a+ 4d+….. a+ (n-1)d) ⇢ (a)

Now lets rewrite the the above equation in reverse order we get the equation as,



Sn= (a+ (n-1)d + a+ (n-2)d+ a+ (n-3)d+ ….. +a) ⇢ (b)

In the next step, add the equation (a) with equation (b), after addition, the result is as follows,

2Sn= (2a+ (n-1)d + 2a+ (n-1)d+…….. + 2a+ (n-1)d) (n terms)

2Sn= [2a + (n-1)d] × d

S_n= \frac{n}{2}[2a+ (n-1)d]

Question 1: Consider the sequence of the first 5 even natural numbers A.P.=2, 4, 6, 8, 10. Find the sum of the A.P.

Solution:

From the above equation, it is known that, a =2, d= 4- 2= 2, and an=10

Using the above equation of sum of n terms in a AP and substituting the values,

Sn= 5/2 [2× 2+ (5-1) × 2]



Sn= 30

So, The sum of the of first 5 even natural numbers is 30

Question 2: Consider the First 5 odd Natural numbers as an A.P. Find the Sum of the series.

Solution:

The first 5 odd Natural numbers are 1, 3, 5, 7, 9

Here, the first term, a= 1, the common difference, d= 3- 1= 2, the number of terms in the series, n=5

S_n= \frac{n}{2}[2a+ (n-1)d]\\S_n= \frac{5}{2}[2(1)+(5-1)2]\\S_n= \frac{5}{2}[10]\\S_n=25

Hence, the sum of the series is 25.

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