Arithmetic Progressions

Introduction

An Arithmetic Progression is a special type of mathemcatical sequence that has varied applications in daily life. It is a special sequence where the consecutive elements vary by a constant difference. In the following module we learn the various concepts associated with Arithmetic Progressions.

What is Arithmetic Progression?

Series are a group of numbers that follow a certain pattern. The most popular mathematical series is the Arithmetic Progression, or AP, which comprises equations that are quite simple to grasp. Two distinct interpretations of AP may be used to comprehend the concept:

• For any pair of successive terms, an Arithmetic Progression, or AP, is a set of numbers wherein the subsequent number may be derived by increasing fixed constant value to the above.

• The general distinction of an Arithmetic Progression of AP is the constant difference that is added to just about any term to generate the next term and its first term.

The First Term and the Common Difference

The generally used components in an Arithmetic Progression, or AP, for a particular series or succession are the first term of AP, its common difference, as well as the nth term.

• Assume the pattern a1, a2, a3, a4…….an is an AP.

Here a1 is said to be the first term.

Using the equation below, we can calculate the constant difference, d:

• d = a2 – a1 = a3 – a2 = a4 – a3…….an – an-1, where d is the common difference, which can be positive,negative, or zero.

The Arithmetic Progression may be represented or stated as follows :

a, a + d, a +2d, a +3d…..a + (n – 1) d, where ‘a’ is an AP’s first term and d is the common difference.

How Do You Calculate the nth Number in An Arithmetic Progression (AP)?

The nth term of an AP(an) is given by

an=a1+(n-1)d

Where a1 is the first term and d is the common difference. This can be intuitively understood from the fact that each term increases by d as the series moves ahead. The common difference may be positive, negative or zero. 

How do you get the sum of an Arithmetic Progression’s initial n terms?

The sum of n terms of an AP can be given by;

S=( n/2) * ( 2a1+(n-1)d)

This is also equal to 

S=(n/2)*(a1+an)  [Implied from the  nth term of the AP]

Arithmetic Progression Formulae

It is critical to know, comprehend, and memorize the equations listed below to solve math equations dependent on an AP’s series and sequential:

• Arithmetic Progression, or AP, in its most standard way:

a, a + d, a + 2d, a + 3d…….a + (n – 1) d

• The nth component of an Arithmetic Progression, or AP, is as follows:

an= a + (n – 1) d

• The sum of n terms of an Arithmetic progression are:

S = n/2*[2a+(n−1) ∗d]

Assuming the very last elements of a continuous Arithmetic Progression or AP are determined, the average of all elements is n/2(a1 + an), whereby ‘a1’ is the first term and ‘an’ would be the last term.

Solved Illustrations

• Question 1:

AP, a = 11, d = 4, and an= 91 in an Arithmetic Progression. Calculate the value of n.

Answer

The first term of the Arithmetic Progression (AP) is 11(a1 = 11), the common difference is 4 (d = 4), and an= 91.

We’re familiar with the equation: an= a1 + (n – 1) d

Let’s see if we can figure out the quantity of ‘n’ by changing the value we possess 

91 = 11 + (n – 1)4

91 = 11 + 4n – 4

91 = 4n + 7

91 – 7 = 4n

84 = 4n

n = 21

Thus 91 is the 21st term of the Arithmetic Progression

• Question 2:

Determine the 30th element in the following Arithmetic Progression:

3, 7, 11, 15,,,,,,,,,,,,,,,,,,

Answer

Let us first find out the first term and common difference

n = 30 (given)

a1 = first term of the AP = 3

d =common difference = difference of two consecutive elements =7-3=11-7= 4

Now as we have seen above:

an= a1 + (n – 1) d

an = 3 + (30 – 1)4

an = 3 + (29)4

an = 3 + 116

an = 119

As a result, the provided AP’s 30th term is 119.