Properties of an Arithmetic Progression

The meaning of the word ‘sequence’ is a collection of objects such that it has known first member, second member, third member, fourth member and so on. A sequence is a function whose domain would be the set of natural numbers N or even a subset thereof. The numbers occurring in a sequence or progression are also known as their terms. 

A sequence of numbers is known as an arithmetic progression if the difference between any two consecutive terms is constant throughout. This difference is called the common difference (c.d.) of the A.P. Let’s learn more about arithmetic progression properties. 

Sequences

The nth term, which we can also write as an, is called the general term of the sequence.

Consider a sequence 2, 4, 6, 8,…

we can obtain each term of the sequence by adding 2 to the previous term. The nth term of this sequence is an = 2n, where n is a natural number.

Series

Let a1, a2, a3, …….., an-1, an be a given sequence, then the expression in the form a₁ + a₂ + a₂ +…+ an is called the series of the above-given sequence. If the given sequence is finite, the series will also be limited. If the given sequence is infinite, the series will also be endless. 

Series are often represented by the Greek letter Σ (sigma). The series a1 + a₂ +…+ an is represented as Σni=1 ai. 

Arithmetic Progression (A.P.)

A sequence of numbers is known as an arithmetic progression if the difference between any two consecutive terms is constant throughout. This difference is called the common difference (c.d.) of the A.P.

In other words, an+1 =an+d, where n belongs to N. Here, a1 is the first term of this arithmetic progression, and the constant term d is a common difference.

For example:

7,5, 3, 1,-1, -3,… are in arithmetic progression. Here the first term is 7, and the common difference is -2.

If we suppose the terms of the given sequence to be a1, a2,…… an. 

They are in A.P. if a2 – a1 = a3 -a2 =…=an – a n-1 = d, where d is the common difference. It is to be noted that equal numbers (zero or non-zero) are in A.P. having a common difference of zero.

Importance of Arithmetic Progression

An arithmetic progression is used to generalise patterns and sequences that we can observe daily. The set of formulas and properties of arithmetic progression help us do this task.

Properties of an Arithmetic Progression

• First arithmetic progression property: 

If any constant is added to each term of an arithmetic progression, the resulting sequence is also an arithmetic progression. In other words, if a1 + a₂ +…+ an  are in A.P., then a1 +k + a₂+k +…+ an +k are also in A.P.

• Second arithmetic progression property: 

If any constant is subtracted from each term of an arithmetic progression, the resulting sequence is also an arithmetic progression. In other words, if a1 + a₂ +…+ an are in A.P., then a1 -k + a₂-k +…+ an -k is also in A.P.

• Third arithmetic progression property: 

If each term of the arithmetic progression is multiplied by a constant, the resulting sequence is also an arithmetic progression. In other words, if a1 + a₂ +…+ an are in A.P., then a1 x k + a₂ x k +…+ an x k are also in A.P.

• Fourth arithmetic progression property: 

If each term of the arithmetic progression is divided by a constant, the resulting sequence is also an arithmetic progression. In other words, if a1 + a₂ +…+ an  are in A.P., then a1 / k + a₂ / k +…+ an  / k are also in A.P.

Arithmetic Progression Questions:

Illustration 1: If a, b, c are in A.P., prove that b + c, c + a, a + b are also in A.P.

Solution:

Given a, b, c are in A.P.

On subtracting a+b+c from each term we get

=> a-(a+b+c), b – (a + b + c) , c – (a + b + c) which is also in A.P. using the second property of arithmetic progression.

=> -(b+c),-(c+a),-(a+b) 

 Therefore, on dividing each term by -1 we get

(b+c),(c+a),(a+b) are in A.P., which is also in A.P. using the fourth property of arithmetic progression.

Conclusion

An arithmetic progression is a series with terms that are consecutive and have a common difference between them as a constant value. An arithmetic progression is used to generalise patterns and sequences that we can observe daily. The set of formulas and properties of arithmetic progression help us do this task. The properties of an arithmetic progression are formed in such a way that it makes it capable enough to handle the maximum number of scenarios.