Arithmetic Progression

In the field of mathematics, Arithmetic Progression, or what is commonly referred to as AP, is a sequence of numbers in a specific or particular order. In our day-to-day lives, we come across quite a few examples of progression, that too, frequently. For instance - the roll numbers of a class, months in a year, days in a week, and so on. 

In mathematics, the pattern of sequences and series has been generalized and is known as progressions. So, let us make ourselves familiar with what is an arithmetic progression along with the terms widely used under this concept, including the first term of the series, common difference, nth term, etc.

What is Arithmetic Progression?

A progression refers to an exclusive type of sequence for which we can find and obtain the formula for the nth term. In mathematics, the most commonly used sequence is that of an Arithmetic Progression or AP and has formulae that are quite easy to understand. The concept of AP can be understood using three different definitions, which are as follows:

• Definition 1: An Arithmetic Progression or AP is a mathematical sequence having a constant difference between two consecutive terms.

• Definition 2: An Arithmetic Progression or AP is a sequence of numbers in which the second number can be obtained by adding a constant or fixed number to the first one for every pair of consecutive terms. 

• Definition 3: The fixed or constant number that is added to any term of an Arithmetic Progression or AP to obtain its next term is called the 'common difference' of AP.

Understanding the Concepts of Common Difference and First Term

In an Arithmetic Progression or AP, for a given series or sequence, the widely used terms include the first term of AP, its common difference, and the nth term. 

Let us suppose that the sequence  \[a_{1}, a_{2}, a_{3}, a_{4},...a_{n}\] is an AP. 

We can obtain the common difference, 'd' using the formula mentioned below:

\[ d = a_{2} - a_{1} = a_{3} - a_{2} =  a_{4} - a_{3} = a_{n} - a_{n-1} \], where 'd' refers to the common difference, and it can be positive, negative, or zero. 

In terms of the common difference, the Arithmetic Progression can be expressed or written as:

\[a, a + d, a +2d, a +3d......a + (n - 1)d\], where 'a' refers to the first term of an AP.

How to find the nth of an Arithmetic Progression or AP?

For finding the nth term of an Arithmetic Progression or AP, the formula is:

an= a + (n - 1)d, where

‘a’ is the first term, d is a common difference, n refers to the number of terms, and an= nth term. 

It is imperative to make a point of the fact that the sequence of an Arithmetic Progression depends on its common difference, that is, d.

If the common difference or d is positive, then the terms of an AP will grow towards the positive side of infinity. On the other hand, if the common difference or d is negative, then the terms of AP will grow towards the negative side of infinity.

How to find the sum of the first n terms of an Arithmetic Progression or AP?

For finding the sum of the first n terms of an Arithmetic Progression or AP, the formula is:

\[ S = \frac{n}{2}(2a + (n-1) \times d)\], where

a is the first term, d is a common difference, n refers to the number of terms, and S is the sum of the first n term of an AP. 

Formulae for Arithmetic Progression 

For solving the mathematical problems based on the series and sequences of an AP, it is essential to know, understand, and learn the formulae specified below:

• General Form of an Arithmetic Progression or AP

a, a + d, a + 2d, a + 3d......a + (n - 1)d

• nth Term of an Arithmetic Progression or AP

an= a + (n - 1)d

• Sum of n Terms of an Arithmetic Progression or AP

\[ S = \frac{n}{2}(2a + (n-1) \times d)\], 

• Sum of all Terms of a Finite Arithmetic Progression or AP, when its first and last terms are known:

\[\frac{n}{2}\](a + l), where 'a' is the first term, and 'l' is the last term.

• Common difference of AP

d = a2- a1 or d=an-an-1 where a1 is the first term and a2 is the second term. Similarly, an is the last term and an-1 is the last but one term. 

Solved Examples on Arithmetic Progression

Question 1

In an Arithmetic Progression or AP, a = 10, d = 5, and an= 95. Find the value of n.

Answer 1

Given – the first term of the Arithmetic Progression or AP is 10 (a = 10), the common difference is 5 (d = 5), and an= 95. 

We know the formula – a_n= a + (n – 1)d

Let us substitute the values we have and determine the value of ‘n’ (number of terms)

95 = 10 + (n – 1)5

95 = 10 + 5n – 5

95 = 5n + 5

95 – 5 = 5n

90 = 5n

n = 18

Hence, the number of terms in this Arithmetic Progression is 18.

Question 2

Find the 20th term of the Arithmetic Progression specified below:

3, 5, 7, 11, 13, 15……….

Answer 2

Let us start by finding the first term and common difference of the given Arithmetic Progression. 

n = 20 (given)

a = first term of the AP = 3

d = common difference = difference between two consecutive terms = 5 – 3 = 2

So, we have to find an. 

an= a + (n – 1)d

an = 3 + (20 – 1)2

an = 3 + (19)2

an = 3 + 38

an = 41

Hence, the 20th term of the given AP is 41. 

Important Points to Remember

• Arithmetic progression is the series where a constant difference is maintained between any two consecutive numbers

• In arithmetic progression ‘a’ is represented as the first term, ‘n’ is represented as the number of terms, ‘d’ is a common difference, Sn is the sum of terms of the AP and tn is the nth term of AP

• Arithmetic progression can be represented as a, a+d, a+2d, a+3d …..

• If a graph is plotted of arithmetic progression then it turns out to be a straight line having slope as the common difference

• The difference in arithmetic progression need not always be positive. It can also take negative values.

This is all about arithmetic progression and its different formulas. Understand the meaning of all the terms used in these formulas and focus on how to use them accordingly.