# How to find the common difference of an Arithmetic Progression whose sum is given?

A **sequence **is defined as the list of items or objects that are arranged in a sequential manner. It is also defined as the set of numbers in a defined order which follows some rule. Suppose if *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}… etc represents the terms of a given sequence, then the subscripts 1, 2, 3, 4… represent the position of the term of the sequence.

There are several types of sequences, some of which are:

- Arithmetic sequence
- Geometric sequence
- Harmonic sequence

The **Arithmetic sequence **is a sequence in which each term is created by the addition or subtraction of a constant number to the preceding number. For example, the series of whole numbers is 0, 1, 2, 3, 4,… is an **arithmetic progression **with a common difference of 1.

In an **arithmetic progression** (**AP**), there are three main terms. These are denoted as:

- Common difference (
*d*) *n*Term (^{th}*a*)_{n}- Sum of the AP (
*S*)_{n}

The difference between the two consecutive numbers is known as the common difference. Consider the numbers *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}…*a _{n} *are in AP, the common difference is calculated by the expression:

d = a_{2 }–a_{1},a_{3 }–a_{2},a_{4 }–a_{3}…,a_{n }–a_{n-1}The AP can also be written in the following form:

a,a+d,a+2d,a+3d,… … …,a+(n-1)dwhere

ais thefirst termof the progression,dis thecommon differenceof the progression anda+(n-1)dis then^{th}term of AP.The sum of the AP is given by

S = n/2[2a + (n – 1)d]

**Steps to calculate the common difference when the** **sum is given:**

The steps to calculate the common difference are given below:

- Substitute the values of sum, the number of terms, and the first term in the formula.
- Simplify the right-hand side.
- Solve for the value of
*d*. - Make sure that the calculation is correct.

### Consider that the first term of an AP is 5, the number of terms in the AP is 9 and the sum of the AP is 189. Calculate the common difference of the above Arithmetic series.

**Solution:**

Given that,

The first term of the AP is 5,

The number of terms in the AP is 9 and

The sum of the AP is 189.

We know that the formula to calculate the sum of the AP is

S2[2_{n }= n/a+ (n-1)d]Substitute 9 for

n, 5 foraand 189 forSinto the formula._{n}189=9/2[2×5+(9-1)

d]42=10+8×

d8

d=32

d=4Hence, the

common differenceof the given series is 4.

**Similar Questions**

**Question 1: Consider that the first term of an AP is 120, the number of terms in the AP is 5 and the sum of the AP is 650. Calculate the common difference of the above Arithmetic series.**

**Solution:**

Given that,

The first term of the AP is 120,

The number of terms in the AP is 5 and

The sum of the AP is 650.

We know that the formula to calculate the sum of the AP is

S

_{n }= n/2[2a + (n-1)d]Substitute 5 for n, 120 for a and 650 for S

_{n}into the formula.650 = 5/2[2 × 120 + (5-1)d]

260 = 240 + 4 × d

4d = 20

d = 5

Hence, the common difference of the given series is 5.

**Question 2:** **Consider that the first term of an AP is 5.5, the number of terms in the AP is 10 and the sum of the AP is 100. Calculate the common difference of the above Arithmetic series.**

**Solution:**

Given that,

The first term of the AP is 5.5,

The number of terms in the AP is 10 and

The sum of the AP is 100.

We know that the formula to calculate the sum of the AP is

S

_{n }= n/2[2a + (n-1)d]Substitute 10 for n, 5.5 for a and 100 for S

_{n}into the formula.100 = 10/2[2 × 5.5 + (10-1)d]

20 = 11 + 9 × d

9d = 9

d = 1

Hence, the common difference of the given series is 1.

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