Arithmetic Progression Formulas: Derivations, Examples - Embibe
• Written By Prince
• Written By Prince

# Arithmetic Progression Formulas: Definition and Examples Arithmetic Progression Formulas: An arithmetic progression (AP) is a sequence in which the differences between each successive term are the same. It is possible to derive a formula for the AP’s nth term from an arithmetic progression. The sequence 2, 6, 10, 14,…, for example, is an arithmetic progression (AP) because it follows a pattern in which each number is obtained by adding 4 to the previous term. In this sequence, the nth term equals 4n-2. The sequence’s terms can be found by substituting n=1,2,3,… in the nth term.

In this article, we will look at the concept of arithmetic progression and the formula for calculating the nth term, common difference, and the sum of n terms of an AP. We will solve various examples based on the arithmetic progression formula to help you better understand the concept.

## What is Arithmetic Progression?

Arithmetic progression is defined as a sequence of numbers, for every pair of consecutive terms, we get the second number by adding a constant to the first one. The constant that must be added to any term of an AP to get the next term is known as the common difference (C.F) of the arithmetic progression.

Arithmetic progression is commonly represented as AP. In an AP, there are 3 main terms that are used to solve mathematical problems:

• (i) Common difference (d)
• (ii) nth Term (an)
• (iii) Sum of the first n terms (Sn)

These three terms define the property of arithmetic progression. We can understand the arithmetic progression concept with an example.

This AP has the first term, a = 2, common difference, d = 4, and last term, l = 50.

This AP has the first term, a = 5, common difference, d = 5, and last term, l = 60.

Get Algebra formulas below:

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The significant formulas associated with arithmetic progression for Class 10th and 12th is listed as follows:

• (i) Sequence
• (ii) Common Difference
• (iii) The nth term of AP (Last term of AP formula)
• (iv) nth term from the last term
• (v) Sum of the first n terms

Let’s see all the formulas in detail.

Formulas for AP

An infinite arithmetic sequence is denoted by the following formula:

Where a represents as the first term and d is the common difference.

• (i) If the value of “d” is positive, then the member terms will grow towards positive infinity.
• (ii)If the value of “d” is negative, then the member terms grow towards negative infinity.

Formula to Calculate Common Difference

The common difference is the fixed constant whose value remains the same throughout the sequence. It is the difference between any two consecutive terms of the AP. The formula for the common difference of an AP is:

Here, an and an+1are two consecutive terms of the AP.

The nth Term of AP Formula

The formula for finding the nth term of an AP is:

Here,

a = First term
d = Common difference
n = Number of terms
an = nth term

Let’s understand this formula with an example:

Example: Find the nth term of AP:
5, 8, 11, 14, 17, …, an, if the number of terms are 12.

Solution:
AP: 5, 8, 11, 14, 17, …, an (Given)
n = 12
By the formula we know, an = a + (n – 1)d
First-term, a = 5
Common difference, d = (8 – 5)= 3
Therefore, an = 5 + (12 – 1)3
= 5 + 33
= 38

Sum of n Terms of AP Formula

For an AP, the sum of the first n terms can be calculated if the first term and the total number of terms are known. The formula for the sum of AP is:

Here,

S = Sum of n terms of AP

n = Total number of terms

a = First term

d = Common difference

Arithmetic Progression Sum Formula When First and Last Terms are Given:

When we know the first and last term of an AP, we can calculate the sum of the arithmetic progressions using this formula:

Derivation:

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

Adding both the equations term wise, we have:

2S = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] + … + [2a + (n – 1) ×d] (n-terms)
2S = n × [2a + (n – 1) × d]
S = n/2[2a + (n − 1) × d]

Let’s understand this formula with examples:

Example 1: Find the sum of the following arithmetic progression:
9, 15, 21, 27, … The total number of terms is 14.
Solution:
AP = 9, 15, 21, 27, …
We have: a = 9,
d = (15 – 9) = 6,
and n = 14
By the AP sum formula, we know:
S = n/2[2a + (n − 1) × d]
= 14/2[2 x 9 + (14 – 1) x 6]
= 14/2[18 + 78]
= 14/2 
= 7 x 96
= 672

Hence, the sum of the AP is 672.

Example 2: Find the sum of the following AP: 15, 19, 23, 27, … , 75.

Solution: AP: 15, 19, 23, 27, … , 75
We have: a = 15,
d = (19 – 15) = 4,
and l = 75

We have to find n. So, using the formula: l = a + (n – 1)d, we get
75 = 15 + (n – 1) x 4
60 = (n – 1) x 4
n – 1 = 15
n = 16

Here the first and last terms are given, so by the AP sum formula, we know:
S = n/2[first term + last term]
Substituting the values, we get:
S = 16/2 [15 + 75]
= 8 x 90
= 720

Hence, the sum of the AP is 720.

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nth Term From the Last Term Formula

When we need to find out the nth term of an AP not from the start but from the last, we use the following formula:

Here,

an = nth term from the last

l = Last term

n = Total number of terms

d = Common difference

### Arithmetic Progression Formulas List

Here we have provided all the arithmetic formulas in the table below for your convenience. Refer to these formulas here or you can also download them as a PDF.

Let’s see some arithmetic progression examples with solutions:

Question 1: The first term of an arithmetic sequence is 4 and the tenth term is 67. What is the common difference?

Solution: Let the first term be a and the common difference d
Use the formula for the nth term: xn = a + d(n − 1)
The first term =
⇒ a = 4 ——- (1)

The tenth term = 67
⇒ x10 = a + d(10 − 1)
= 67
⇒ a + 9d = 67 ——- (2)

Substitute a = 4 from (1) into (2)
⇒ 4 + 9d = 67
⇒ 9d = 63
⇒ d = 63 ÷ 9
= 7

The common difference is 7.

Question 2: What is the thirty-second term of the arithmetic sequence -12, -7, -2, 3, … ?

Solution: This sequence has a difference of 5 between each pair of numbers.
The values of a and d are:
a = -12 (the first term)
d = 5 (the “common difference”)
The rule can be calculated:
xn = a + d(n − 1)
= -12 + 5(n − 1)
= -12 + 5n − 5
= 5n − 17

So, the 32nd term is:
x32 = 5 × 32 − 17
= 160 − 17
= 143

Question 3: What is the twentieth term of the arithmetic sequence 21, 18, 15, 12, … ?

Solution: This sequence is descending, so has a difference of -3 between each pair of numbers.
The values of a and d are:
a = 21 (the first term)
d = -3 (the “common difference”)

The rule can be calculated:
xn = a + d(n-1)
= 21 + -3(n-1)
= 21 – 3n + 3
= 24 – 3n

So, the 20th term is:
x20 = 24 – 3 × 20
= 24 – 60
= -36

Question 4: What is the sum of the first thirty terms of the arithmetic sequence: 50, 45, 40, 35, … ?

Solution: 50, 45, 40, 35, …
The values of a, d and n are:
a = 50 (the first term)
d = -5 (common difference)
n = 30 (how many terms to add up)

Using the sum of AP formula – Sn = n/2(2a + (n – 1)d), we get:
S30 = 30/2(2 × 50 + 29 × -5))
=15(100 – 145)
= 15 × -45
= -675

Question 5: What is the sum of the eleventh to twentieth terms (inclusive)of the arithmetic sequence: 7, 12, 17, 22, …?

Solution: Given AP: 7, 12, 17, 22, …
The values of a and d:
a = 7 (first term)
d = 5 (common difference)
To find the sum of the eleventh to twentieth terms we subtract the sum of the first ten terms from the sum of the first 20 terms

Therefore the sum of the eleventh to twentieth terms = 1,090 – 295
= 795

Other important Maths concepts:

### Arithmetic Progression Problems

Here are some arithmetic progression questions for you to practice.

Question 1: What is the seventh term of the arithmetic progression 2, 7, 12, 17, …?

Question 2: What is the sum of the first 50 odd positive integers?

Question 3: 13 + 28 + 43 + ⋯ + ​an = 68210
The n terms being added on the left side of the above equation form an arithmetic progression in that order. What is n?

Question 4: Consider an arithmetic progression whose first term and common difference are both 100. If the nth term of this progression is equal to 100!, find n.

Question 5: You are standing next to a bucket and are tasked with collecting 100 potatoes, but can only carry one potato at a time. The potatoes are in a line in front of you, with the first potato 1 meter away and each subsequent potato is located an additional one meter away. How much distance would you cover while performing this task?

Question 6: Solve the following expression:
(100001 + 100003 + 100005+ ⋯ + 199999​)/ (1 + 3 + 5 + 7 + ⋯ + 99999) = ?

Question 7: For a certain arithmetic progression with S1729 = S29, where Sn denotes the sum of the first n terms, find S1758.

Question 8: Sunil got −10 marks in his first exam and 15 marks in his 15th exam. If all his marks follow an arithmetic progression with a positive common difference, in which exam did he get zero marks?

Also, Check

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### FAQs on Arithmetic Progression

Here we have provided some of the frequently asked questions:

Q.1: What is the sum of the first n natural numbers?
Ans: With the help of AP sum formula, we can calculate the sum of the first n natural numbers.
S = n(n + 1)/2

Q.2: What is the sum of first n even numbers?
Ans: Let the sum of first n even numbers is Sn
Sn = 2+4+6+8+10+…………………..+(2n)
Solving the equation using the AP sum formula, we get:
Sum of n even numbers = n(n + 1)

Q.3: How many formulas are there in arithmetic progression Class 10?
Ans: There are mainly two formulas associated with arithmetic progression:
(i) nth term of an AP
(ii) Sum of n terms of an AP

Q.4: What is arithmetic progression?
Ans: Arithmetic progression is defined as a sequence of numbers in which each number differs from the preceding one by a constant quantity (known as common difference).

Q.5: What is the Arithmetic Progression Formula?
Ans: The arithmetic sequence is given by a, a + d, a + 2d, a + 3d, … . Hence, the formula to find the nth term is:
an = a + (n – 1) × d.
Sum of n terms of the AP = n/2[2a + (n − 1) × d].

Q.6: What is d in AP formula?
Ans: d is the common difference. Arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term.

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Now you are provided with all the necessary information regarding arithmetic progression formulas. We hope that you have downloaded the PDF of AP formulas available on this page. Practice more questions and master this concept.

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