# How to find the common difference of an Arithmetic Progression?

A sequence is also known as a progression and is defined as the successive arrangement of numbers in order while following some specific rules. Depending upon the set of rules followed by a sequence, it is classified into various kinds, such as an arithmetic sequence, geometric sequence, harmonic sequence, and Fibonacci sequence. An arithmetic sequence or progression is a sequence of numbers when the difference between any two successive numbers is the same. For instance, 5, 10, 15, 20, 25, 30,… is an arithmetic sequence, where the common difference between any two successive terms is 5.

**The general form of an arithmetic progression is,**

AP = a, a + d, a + 2d, a + 3d,……., a + (n-1) dWhere,

“a”is the first term of the progression, and

“d”is a common difference.

### Common Difference

The term **“common difference”** is a concept that is used in sequences and arithmetic progressions. A common difference is a difference between any term and its preceding term in an arithmetic sequence, and it is generally represented by the letter “d”. So, to determine the common difference of an arithmetic sequence, subtract the first term from the second term, the second term from the third term, etc. The common difference of an arithmetic sequence plays a vital role in determining the successive terms of the sequence.

For instance, 3, 7, 11, 15, 19, 23,… is an arithmetic sequence. The common difference between any two successive terms of a given sequence is 4, i.e.,

- Second term – First term = 7 – 3 = 4
- Third term – Second term = 11 – 7 = 4
- Fourth term – Third term = 15 – 11 = 4
- Fifth term – Fourth term = 19 – 15 = 4, and so on.

There are two types of arithmetic progression based on the common difference, i.e.,

**An arithmetic progression increases if the common difference is positive, and an arithmetic progression decreases if the common difference is negative. **

For example, 2, 4, 6, 8, 10, 12,… is an increasing arithmetic progression as the common difference is positive, i.e., “2”. 10, 4, -2, -8, -14,… is a decreasing arithmetic progression as the common difference is negative, i.e., “-6”.

Note: Common difference of an arithmetic sequence remains the same if a constant quantity is added or subtracted to or from each term of the arithmetic sequence.

For example, 4, 9, 14, 19, 24, 29,… is an arithmetic progression with a common difference of “5”. If “7” is added to each term of the given progression, then it becomes 11, 16, 21, 26, 31, 26,… and the common difference between the successive terms of the new sequence is also “5”. If “2” is subtracted from each term of the given progression, then it becomes 2, 7, 12, 17, 22, 27,… and the common difference between the successive terms of the new sequence is also “5”.

### Formula of common differences of an A.P

So, the formula for finding the common difference is,

**Formula of common difference if the sequence is given,**

Let the sequence be, a_{1}, a_{2}, a_{3},……,a_{n-1}, an

Now, the common difference in the sequence is

d = a_{2} – a_{1} = a_{3} – a_{2} = …… = a_{n} – a_{n-1}

Hence,

d = a_{n}-a_{n-1}where,

ais the nth term and_{n}

ais its preceding term._{n-1}

**Formula of common difference when the nth term and the first term of the sequence are given,**

Let “a” be the first term and “d” be the common difference in the arithmetic sequence. Now, the nth term of the sequence is,

a_{n} = a + (n-1) d

By subtracting the first term from the nth term, we get

⇒ a_{n} – a_{1 }= a + (n–1) d – a

⇒ a_{n} – a_{1} = (n–1)d

⇒ d = (a_{n} – a_{1})/(n – 1)

Hence,

d = (a_{n}-a_{1})/(n-1)where,

ais the nth term and_{n}

ais the first term._{1}

**Formula of common difference when the sum of n terms and the first term of the sequence are given,**

Let “a” be the first term and “d” be the common difference in the arithmetic sequence.

The sum of n terms of an arithmetic sequence is,

S_{n} = (n/2)[2a + (n-1) d]

⇒ S_{n} × (2/n) = 2a + (n-1)d

⇒ (S_{n }× 2/n) – 2a = (n-1)d

⇒ d = [(S_{n} × 2/n) – 2a]/(n-1)

Hence,

d = [(S_{n}×2/n) – 2a_{1})]/(n-1)where,

Sis the sum of n terms and_{n}

ais the first term._{1}

### Sample Problems

**Problem 1: Determine the common difference in the given sequence: 20, 16, 12, 8, 4, 0, -4,…**

**Solution:**

Given sequence: 20, 16, 12, 8, 4, 0, -4,…

We know that a common difference is difference between any term and its preceding term in an arithmetic sequence, i.e.,

d = a

_{n}– a_{n-1}, where a_{n}is the nth term and a_{n-1}is its preceding term.Here, a

_{1}= 20, a_{2}= 16, a_{3}= 12, a_{4}= 8, a_{5}= 4, a_{6}= 0, a_{7}= -4,…d = a

_{4}– a_{3}= 8 – 12 = -4d = a

_{3}– a_{2}= 12 – 16 = -4Hence, the common difference in the given sequence is -4.

**Problem 2: Determine the common difference of a sequence of 10 terms whose sum is 120 and the first term is 6.**

**Solution:**

Given,

Number of terms of a sequence (n) = 10

sum of the 10 terms (S

_{10})= 120First-term (a) = 6

We know that the sum of n terms of an arithmetic sequence is,

S

_{n}= (n/2)[2a + (n-1)d]⇒ 120 = (10/2)[6 + (10 -1)d]

⇒ 120 = 5[6 + 9d]

⇒ (6 + 9d) = 120/5 = 24

⇒ 9d = 24 – 6 = 18

⇒ d = 18/9 = 2

Hence, the common difference in the given sequence is 2.

**Problem 3: Three terms are in A.P. Determine the common difference if the first and third terms are 17 and 23, respectively.**

**Solution:**

Given,

Let the three terms of the arithmetic sequence be a, a + d, a+ 2d.

The first term (a) = 17, and

The third term (a + 2d) = 23

Now, third term – first term = 23 – 17

⇒ a + 2d – a = 6

⇒ 2d = 6

⇒ d = 6/2 = 3

Hence, the common difference in the given sequence is 3.

**Problem 4: How to find the common difference in an arithmetic sequence?**

**Solution:**

A common difference is a difference between any term and its preceding term in an arithmetic sequence. So, to determine the common difference of an arithmetic sequence, subtract the first term from the second term, the second term from the third term, etc.

So, the formula for finding the common difference is,

d = a_{n}-a_{n-1},where

ais the nth term and_{n}

ais its preceding term._{n-1}An A.P increases if the common difference is positive, and an A.P decreases if the common difference is positive.

**Problem 5: Determine the common difference if the first and eighth terms of an arithmetic sequence are 12 and 68, respectively.**

**Solution:**

Given,

The first term (a) = 12

The eighth term = 68

We know that,

The nth term of an A.P. = a + (n-1)d

eighth term = a + (8-1)d = a + 7d

Now, eighth term – first term = 68 – 12

a + 7d – a = 56

⇒ 7d = 56

⇒ d = 56/7 = 8

Hence, the common difference in the given sequence is 8.

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