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Arithmetic progression

An arithmetic progression or arithmetic sequence is a number sequence in which the common difference between successive terms is constant. In other words, after adding a common constant number to any term, we would get the successive term and so on.

A progression is a pattern-following sequence of integers. It is a type of sequence for which the formula for the nth term can be obtained.

There are generally three types of progressions in mathematics. They are as follows:

Geometric Progression (GP)

Arithmetic progression (AP)

Harmonic Progression (HP) 

What is arithmetic progression?

The arithmetic progression is the most often used sequence in mathematics, with simple formulas. An AP is a sequence in which the differences between each successive term are the same. It is possible to obtain a formula for the nth term from an arithmetic progression. 

The series ‘1, 5, 9, 13,…’ is an arithmetic progression (AP) because it trails a fashion in which each number is obtained by adding 4 to its proceeding term. In this series, the nth term equals 4n-3. The sequence’s terms can be derived by substituting n = 1,2,3,… in the nth term.

When n = 1, 4n-3 = 4(1) – 3 = 4-3= 1

When n = 2, 4n-3 = 4(2) – 3 = 8-3 = 5

When n = 3, 4n-3 = 4(3)-3 = 12 – 3 = 9

Arithmetic progression has three types of definitions.

Definition 1: An abbreviation for a mathematical sequence in which the difference between two consecutive words is always a constant.

Definition 2: An arithmetic progression is a number sequence in which the second number is obtained by adding a fixed number to the first one for every pair of successive terms.

Definition 3: The common difference of an arithmetic progression is the set number that must be added to any term of an AP to get the next term. Consider the series ‘1, 4, 7, 10, 13, 16,…’ which is an arithmetic sequence with a common difference of 3.

Arithmetic Progression Terminology

Here are some more examples of arithmetic progressions:

5,11 , 17, 23, 29,…

93, 84, 75, 66, 57,…

3, 8,13,18,…

An AP is typically represented as follows: a1, a2, a3,…

First Term: The first term of an AP is the progression’s first number. It is typically denoted by a1 (or) a. For example, the first term in the sequence 5, 11, 17, 23, 29,… is 5. That is, a1 = 5 (or) a = 5.

Common Difference: As we already know that in an A.P, all the terms after the first one are found by adding some particular constant d to the previous term. This constant d is called the common difference. If the first term is a1, the second term is a1+d, the third term is a1+2d, the fourth term is a1+3d, and so on. In the series ‘5, 11, 17, 23, 29,…’ each term, except the first, is created by adding 6 to the previous term.

As a result, the common difference is d = 6. In general, the common difference is the difference between 2 consecutive terms of the AP. As a result, the formula for computing an AP’s common difference is: d = a(n) – a(n – 1).

In general an arithmetic sequence can be written like: {a, a+d, a+2d, a+3d, … }

Using the above example, we get: {a, a+d, a+2d, a+3d, … } = {5, 5+6, 5+2×6, 5+3×6, … } = {5, 11, 17, 23,….}

General Term of an arithmetic progression (nth term)

The formula an = a + (n-1)d calculates the general term or nth term of an AP whose first term is a and the common difference is d. For example, to find the general term or nth term of the sequence ‘6, 13, 20, 27, 34,….’ we use the formula for the nth term and substitute the value of the initial term, a1 = 6, and the common difference, d = 7.

The result is an = a + (n-1)d = 6 + (n-1)7 = 6 + 7n-7 = 7n-1 

As a result, a = 7n-1 is the general term or nth term of this series. 

Let’s discuss the purpose of locating an AP’s general term. We know that we can surely find a term by appending d to its prior term. For example, if we need to find the 6th term in the above series, we can simply add d = 7 to the 5th term, which is 34. 

6th term = 5th term + 7 = 34+7 = 41

What if we need to look up the 102nd term? Isn’t it hard to do it by hand? In this example, we can simply substitute n =  102 (as well as a = 6 and d = 7) in the calculation for an AP’s nth term. 

an = a + (n-1)d 

a102 = 6 + (102-1)7

a102 = 6 + (101)7 

a102 = 713

As a result, the 102nd term in the preceding sequence is 713. As a result, the general term (or) nth term of an AP is used to find any term of the AP without first locating its prior term.

Formula Lists

The formulas used in AP are given in a tabular form. 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

an = a + (n – 1) × d

Sum of n terms in AP

S = n2 [2a + (n − 1) × d]

Sum of all terms in a finite AP with the last term as ‘l’

n2(a + l)

Conclusion 

  • An AP is a set of numbers in which each phrase is obtained by adding a fixed number to the number before it.

  • a is the first term, d is a common difference, an is the nth term, and n is the total number of terms.

  • In general, AP can be written as a, a+d, a+2d, a+3d,…. and the nth term of an AP can be calculated as an = a + (n-1)d.

  • The sum of an AP can be calculated as sn = n2 [2a+(n1)d]

  • An AP graph is a straight line with the slope serving as the common difference.