Progressions

Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value. It is also termed an Arithmetic Sequence. For example, the series of natural numbers: 1, 2, 3, 4, 5, 6,… is an Arithmetic Progression, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1). Even in the case of odd numbers and even numbers, we can see the common difference between two successive terms will be equal to 2.

If we observe in our regular lives, we come across Arithmetic progression quite often. For example, Roll numbers of students in a class, days in a week or months in a year. This pattern of series and sequences has been generalized in Maths as progressions.

Also Read: How to Prepare for CSAT Reasoning

Finding the sum of the given number of terms in an arithmetic progression

Let a denote the first term d, the common difference, and n the total number of terms. Also, let L denote the last term, and S the required sum; then

S= n(a+L)/2
L = a + (n – 1) d
S = n/2 [2a+(n–1) d]

Arithmetic Mean

Let, a and b be two quantities and A be their arithmetic mean.

A=(a+b)/2

Geometric Progression

Quantities exhibit Geometric Progression characteristics when they are altered by a consistent multiplier. This multiplier is also known as the common ratio, and its determination involves dividing any term by the term directly preceding it.
If we examine the series a, ar, ar2 , ar3 , ar4 ,…
we notice that in any term the index of r is always less by one than the number of the term in the series.

Sum of terms in GP

Let a be the first term, r the common ratio, n the number of terms, and Sn be the sum to n terms.

Sn=a(r^n-1)/(r-1) If r > 1, then

Sn=a({1-r}^n )/(1-r) If r< 1, then

Geometric Mean

Let a and b be the two quantities; G the geometric mean.

G=√ab

Harmonic Progression

If a, b, c, d is in A.P. then 1/a, 1/b, 1/c and 1/d are all in H.P.

What is Arithmetic Progression?

In mathematics, there are three different types of progressions. They are:

Arithmetic Progression (AP)
Geometric Progression (GP)
Harmonic Progression (HP)

A progression is a special type of sequence for which it is possible to obtain a formula for the nth term. The Arithmetic Progression is the most commonly used sequence in maths with easy to understand formulas.

Definition 1: A mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP.

Definition 2: A sequence of numbers is termed an arithmetic sequence or progression when the second number in each consecutive pair is obtained by adding a constant value to the first number.

The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP. Now, let us consider the sequence, 1, 4, 7, 10, 13, 16,…

It is considered as an arithmetic sequence (progression) with a common difference 3.

Notation in Arithmetic Progression

In AP, we will come across some main terms, which are denoted as:

First term (a)
Common difference (d)
nth Term (a_n)
Sum of the first n terms (S_n)

All three terms represent the property of Arithmetic Progression. We will learn more about these three properties in the next section.

First Term of AP

The AP can also be written in terms of common differences, as follows;

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

where “a” is the first term of the progression.

Common Difference in Arithmetic Progression

In this progression, for a given series, the terms used are the first term, the common difference and nth term. Suppose, a₁, a₂, a₃, ……………., a_n is an AP, then;the common difference “ d ” can be obtained as;

d = a₂ – a₁ = a₃ – a₂ = ……. = a_n – a_{n – 1}

Where “d” is a common difference. It can be positive, negative or zero.

General Form of an AP

Consider an AP to be: a₁, a₂, a₃, ……………., a_n

Position of Terms	Representation of Terms	Values of Term
1	a₁	a = a + (1-1) d
2	a₂	a + d = a + (2-1) d
3	a₃	a + 2d = a + (3-1) d
4	a₄	a + 3d = a + (4-1) d
.	.	.
.	.	.
.	.	.
.	.	.
n	a_n	a + (n-1)d

Arithmetic Progression Formulas

There are two major formulas we come across when we learn about Arithmetic Progression, which is related to:

The nth term of AP
Sum of the first n terms

Let us learn here both the formulas with examples.

nth Term of an AP

The formula for finding the n-th term of an AP is:

a_n = a + (n − 1) × d

Where

a = First term

d = Common difference

n = number of terms

a_n = nth term

Example: Find the nth term of AP: 1, 2, 3, 4, 5…., a_n, if the number of terms are 15.

Solution: Given, AP: 1, 2, 3, 4, 5…., a_n

n=15

By the formula we know, a_n = a+(n-1)d

First-term, a =1

Common difference, d=2-1 =1

Therefore, a_n = a₁₅ = 1+(15-1)1 = 1+14 = 15

Note: The behaviour of the sequence depends on the value of a common difference.

If the value of “d” is positive, then the member terms will grow towards positive infinity
If the value of “d” is negative, then the member terms grow towards negative infinity

Types of AP

Finite AP: An AP containing a finite number of terms is called finite AP. A finite AP has a last term.

For example: 3,5,7,9,11,13,15,17,19,21

Infinite AP: An AP which does not have a finite number of terms is called infinite AP. Such APs do not have a last term.

For example: 5,10,15,20,25,30, 35,40,45………………

Sum of N Terms of AP

“The formula for calculating the sum of the first n terms of an arithmetic progression (AP) can be explained when you know the first term, common difference, and the total number of terms. This formula for the sum of an arithmetic progression is detailed below:”

Consider an AP consisting “n” terms.

S_n = n/2[2a + (n − 1) × d]

This is the AP sum formula to find the sum of n terms in series.

Proof: 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 have:

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

Adding both the equations term wise, we have:

2S_n = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] + …………. + [2a + (n – 1) ×d] (n-terms)

2S_n = n × [2a + (n – 1) × d]

S_n = n/2[2a + (n − 1) × d]

Example: Let us take the example of adding natural numbers up to 15 numbers.

AP = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Given, a = 1, d = 2-1 = 1 and a_n = 15

Now, by the formula we know;

S_n = n/2[2a + (n − 1) × d]

S₁₅ = 15/2[2.1+(15-1).1]

= 15/2[2+14]

= 15/2 [16]

= 15 x 8

= 120

Hence, the sum of the first 15 natural numbers is 120.

Sum of AP when the Last Term is Given

Find the sum of an arithmetic progression (AP) when the first and last terms are provided.”

S = n/2 (first term + last term)

List of Arithmetic Progression Formulas

The list of formulas is given in a tabular form used in AP. These formulas are useful to solve problems based on the series and sequence concept.

General Form of AP	a, a + d, a + 2d, a + 3d, . . .
The nth term of AP	a_n = a + (n – 1) × d
Sum of n terms in AP	S = n/2[2a + (n − 1) × d]
Sum of all terms in a finite AP with the last term as ‘l’	n/2(a + l)

Arithmetic Progressions Solved Examples

“Below, we have solved the following problems that detail the process of finding the nth term and the sum of the sequence using AP sum formulas.” Go through them once and solve the practice problems to excel in your skills.

Example 1: Find the value of n, if a = 10, d = 5, a_n = 95.

Solution: Given, a = 10, d = 5, a_n = 95

From the formula of general term, we have:

a_n = a + (n − 1) × d

95 = 10 + (n − 1) × 5

(n − 1) × 5 = 95 – 10 = 85

(n − 1) = 85/ 5

(n − 1) = 17

n = 17 + 1

n = 18

Example 2: Find the 20th term for the given AP:3, 5, 7, 9, ……

Solution: Given,