Arithmetic Progression Formulas

There are two major formulas we come across when we learn about Arithmetic Progression, which is related to:

• The nth term of AP
• Sum of the first n terms

Let us learn here both the formulas with examples.

nth Term of an AP

The formula for finding the n-th term of an AP is:

an = a + (n − 1) × d

Where 

a = First term

d = Common difference

n = number of terms

a_n = nth term

Example: Find the nth term of AP: 1, 2, 3, 4, 5…., a_n, if the number of terms are 15.

Solution: Given, AP: 1, 2, 3, 4, 5…., a_n

n=15

By the formula we know, a_n = a+(n-1)d

First-term, a =1

Common difference, d=2-1 =1

Therefore, a_n = a₁₅ = 1+(15-1)1 = 1+14 = 15

Note: The behaviour of the sequence depends on the value of a common difference.

• If the value of “d” is positive, then the member terms will grow towards positive infinity
• If the value of “d” is negative, then the member terms grow towards negative infinity

Types of AP

Finite AP: An AP containing a finite number of terms is called finite AP. A finite AP has a last term.

For example: 3,5,7,9,11,13,15,17,19,21

Infinite AP: An AP which does not have a finite number of terms is called infinite AP. Such APs do not have a last term.

For example: 5,10,15,20,25,30, 35,40,45………………  

Sum of N Terms of AP

For an AP, the sum of the first n terms can be calculated if the first term, common difference and the total terms are known. The formula for the arithmetic progression sum is explained below:

Consider an AP consisting “n” terms.

Sn = n/2[2a + (n − 1) × d]

This is the AP sum formula to find the sum of n terms in series.

Proof: Consider an AP consisting “n” terms having the sequence a, a + d, a + 2d, …………., a + (n – 1) × d

Sum of first n terms = a + (a + d) + (a + 2d) + ………. + [a + (n – 1) × d] ——————-(i)

Writing the terms in reverse order,we have:

S_n= [a + (n – 1) × d] + [a + (n – 2) × d] + [a + (n – 3) × d] + ……. (a) ———–(ii)

Adding both the equations term wise, we have:

2S_n = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] + …………. + [2a + (n – 1) ×d] (n-terms)

2S_n = n × [2a + (n – 1) × d]

S_n = n/2[2a + (n − 1) × d]

Example: Let us take the example of adding natural numbers up to 15 numbers.

AP = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Given, a = 1, d = 2-1 = 1 and a_n = 15

Now, by the formula we know;

S_n = n/2[2a + (n − 1) × d]

S₁₅ = 15/2[2.1+(15-1).1]

= 15/2[2+14]

= 15/2 [16]

= 15 x 8

= 120

Hence, the sum of the first 15 natural numbers is 120.

Sum of AP when the Last Term is Given

Formula to find the sum of AP when first and last terms are given as follows:

S  = n/2 (first term + last term)

List of Arithmetic Progression Formulas

The list of formulas is given in a tabular form used in AP. These formulas are useful to solve problems based on the series and sequence concept.

General Form of AP	a, a + d, a + 2d, a + 3d, . . .
The nth term of AP	an = a + (n – 1) × d
Sum of n terms in AP	S = n/2[2a + (n − 1) × d]
Sum of all terms in a finite AP with the last term as ‘l’	n/2(a + l)

Arithmetic Progressions Solved Examples

Below are the problems to find the nth term and the sum of the sequence, which are solved using AP sum formulas in detail. Go through them once and solve the practice problems to excel in your skills.

Example 1: Find the value of n, if a = 10, d = 5, a_n = 95.

Solution: Given, a = 10, d = 5, a_n = 95

From the formula of general term, we have:

a_n = a + (n − 1) × d

95 = 10 + (n − 1) × 5

(n − 1) × 5 = 95 – 10 = 85

(n − 1) = 85/ 5

(n − 1) = 17

n = 17 + 1

n = 18

Example 2: Find the 20th term for the given AP:3, 5, 7, 9, ……

Solution: Given, 

3, 5, 7, 9, ……

a = 3, d = 5 – 3 = 2, n = 20

a_n = a + (n − 1) × d

a₂₀ = 3 + (20 − 1) × 2

a₂₀ = 3 + 38

⇒a₂₀ = 41

Example 3: Find the sum of the first 30 multiples of 4.

Solution:

The first 30 multiples of 4 are: 4, 8, 12, ….., 120

Here, a = 4, n = 30, d = 4

We know,

S₃₀ = n/2 [2a + (n − 1) × d]

S₃₀ = 30/2[2 (4) + (30 − 1) × 4]

S₃₀ = 15[8 + 116]

S₃₀ = 1860

Practice Problems on AP

Find the below questions based on Arithmetic sequence formulas and solve them for good practice.

Question 1: Find the a_n and 10th term of the progression: 3, 1, 17, 24, ……

Question 2: If a = 2, d = 3 and n = 90. Find a_n  and S_n.

Question 3: The 7th term and 10th terms of an AP are 12 and 25. Find the 12th term.