Arithmetic Progression Formula: Everything You Need to Know

# Arithmetic Progression Formula: Everything You Need to Know

Last updated:
9th Feb, 2021
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## Introduction

An arithmetic progression is a sequence in which the next term in the sequence is obtained by adding a constant to each term. The constant added is called the common difference. It is a sequence such that the difference between any two consecutive terms in the sequence is always a constant.

Suppose, n1, n2, n3……..nn are the

terms of an arithmetic progression sequence.

Then, n2 = n1 + d, n3 = n2 + d and so on.

Where n1 = the first term and d is the common difference

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## Arithmetic Progression Examples

Verify if the following sequence 3, 6, 9, 12, 15 is an arithmetic progression or not.
For this sequence to be an arithmetic progression sequence, the common difference between the consecutive terms should be constant.

Common difference (d) = n2 – n1 must equal to n3 – n2 and so on.

In this sequence, d = 6 – 3 = 3, 9 – 6 = 3, 12 – 9 = 3, and 15 – 12 = 3.

The difference between consecutive terms is constant. Hence, the above sequence is an arithmetic progression.

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## Arithmetic Progression Formula

To understand the arithmetic progression formula, one should be familiar with the terminologies used in the formula.

First-term

As the name states, the first term is the first term of the sequence, which is usually represented by n1. For example, in the 5, 12, 19, 26, 33 sequence, the first term is 5.

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Common Difference

A common difference is the fixed number that is added or subtracted between two consecutive terms (except the first term) in the arithmetic progression. It is denoted by ‘d’.

For example, if n1 is the first term, then:

n2 = n1 + d

n3 = n2 + d and so on

### Arithmetic Progression Formula to Find theGeneral Term or nth Term

The general term or nth term in an arithmetic progression is found by:

Nn = a + (n-1) *d

where ‘a’ is the first term and ‘d’ is a common difference.

So, 1st term, N1 = a + (1-1) *d

2nd term, N2 = a + (2-1) *d

3rd term, N3 = a + (3-1) *d

By computing ‘n’ terms in the above formula, we get the general form of an arithmetic progression.

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

### Arithmetic Progression Formula to Find theSum

The arithmetic progression formula for the sum of ‘n’ terms where ‘a’ is the first term and ‘d’ is a common difference is as follows.

When the nth term is unknown:

Sn = (n/2) * [2a + (n − 1) * d]

When the nth term is known:

Sn = (n/2) * [a1 + an]

Formula derivation

Let us assume that ‘t’ is the nth term of the series and Sn is the sum of first n terms in an arithmetic progression:  a, (a + d), (a + 2d), …., a + (n – 1) * d.

Then,

Sn = a1 + a2 + a3 + ….an-1 + an

Substituting the terms in the above formula, we get

Sn = a + (a + d) + (a + 2d) + …….. + (t – 2d) + (t – d) + t                  …(1)

After writing the equation (1) in the reverse order

Sn =t + (t – d) + (t – 2d) + …….. + (a + 2d) + (a + d) + a                  …(2)

Now, add equation (1) and (2), we get

2Sn = (a + t) + (a + t) + (a + t) + …….. + (a + t) + (a + t) + (a + t)

2Sn = n * (a + t)

Sn = (n/2) * (a + t)                                                                               …(3)

Let us replace the last term ‘t’ by the nth term in equation 3, we get,

nth term = a + (n – 1) * d

Sn = (n/2) * {a + a + (n – 1) * d}

Sn = (n/2) * {2a + (n – 1) * d}

## Example

If you are asked to find the sum of the first 30 terms of a sequence 5, 11, 17, 23, ……

Solution:

a = 5, d = a2 – a1 = 11 – 5 = 6

Sn = (n/2) * {2a + (n – 1) * d}

Sn = (30/2) * (2 * 5 + (35 – 1) * 6}

Sn = (15) * (10 + 204)

Sn = 15 * 214

Sn = 3210

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## Conclusion

In mathematics, an arithmetic progression is a series of numbers where the difference between two consecutive terms is always constant. We can find multiple examples of arithmetic progression in our daily life. For example, enrollment numbers of students in a batch, months in a year, etc.

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Director of Engineering @ upGrad. Motivated to leverage technology to solve problems. Seasoned leader for startups and fast moving orgs. Working on solving problems of scale and long term technology strategy.
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1What are the different types of progressions in mathematics?

Numbers are sorted in a predictable order when they are arranged in a progression. Progressions have the ability to anticipate the next numbers in a series in a given set of integers. There are three different types of progressions that are used in mathematics, namely, arithmetic progression (AP), harmonic progression (HP) and geometric progression (GP). In AP, the common difference is used to find the next term, in GP, common ratio is used while HP basically means that the reciprocal of the given terms are in AP.

2What are the two types of arithmetic progression series?

There are two types of arithmetic progression series in mathematics- finite series and infinite series. In finite series, the number of terms are either known or at least it is given that they are limited. While in an infinite sequence, the number of terms are infinite. For finding the common difference, the formula is the same for both the arithmetic progression series. But when it comes to finding the sum, the formula differs.

3How is an arithmetic progression related to harmonic progression?

In an arithmetic progression, the common difference is taken out, and then, using the first term and the common difference, the sum of the series is calculated. When it comes to harmonic progression, there isn’t any difference between finding the common difference and the sum of the series. The terms of the given HP are reciprocated, and then the same formula as AP is used. Thus, when the terms of the HP are reciprocated, the series becomes an AP. That is how AP and HP are connected.

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