Properties Of Arithmetic Progression

What is arithmetic progression (AP)?

AP is said to be a sequence of series. Moreover, if we add or substrate a constant number in each term, we will get the same difference. 

For instance, the series 11,13,14,16,18 is an arithmetic progression, as the difference between terms is the same, i.e. 2.

However, a series like 11,13,15,18,21 is not an arithmetic progression; it will vary if we make the difference between these terms. For instance,

11 – 13 = 2

13 – 15 = 2

15 – 18 = 3

18 – 21 = 3

So, the differences between the terms are not constant. As a result, it is not an arithmetic progression.

‘Arithmetic’ means there are three numbers. For instance, if we have c, B, and d, then B is called the arithmetic mean of the numbers c and d.

Questions related to average arithmetic mean are generally asked in the exams. Moreover, notes on average arithmetic mean play an important role in preparation for the exams.

Properties of Arithmetic Progression

• Property 1:

If we add or subtract a constant from each term of AP, then we will get the resulting sequence as an AP only. Moreover, the difference between the terms should be the same or common. Only then it can be called an arithmetic progression.

Proof:

Let a1, a2, a3, and so on be an AP; the difference is common and is denoted by d. And let the fixed constant be m, which is added to each of the terms of AP. Then the sequence will come as:

a1 + m, a2+ m, a3 + m…

Let bn = an + m, n = 1,2…, then the new sequence will come as b1, b2, b3,…

We have, bn + 1 – bn = (an+1  + m) – (an + m)

an + 1 – an = d for all n belongs to N

• Property 2

If we multiply or divide each term of AP by a non-zero constant m, then AP will be the resulting sequence only. Moreover, md or d/m is a common difference, whereas the given AP’s common difference is d.

Proof:

Let a1, a2, a3,… be an AP, with d as a common difference, and let the non-zero constant be m. Let b1, b2, b3,… be the sequence got by multiplying each term of given AP by m, then.

b1 = a1 m, b2 = a2 m ,…, bn = an m

Now, bn + 1 – bn = an + 1 m – an m

(an + 1 – an)m

= dm for all n belong to N

Thus, the new sequence is an arithmetic progression, with dm as a common difference.

• Property 3

In a finite arithmetic progression, the sum of the terms equidistant from beginning to end is always the same. Moreover, it is equal to the sum of the first and last term i.e. ak + an – (k – 1) = a1 + an for all k = 1, 2, 3,…, n-1.

• Property 4

If the nth term is a linear expression in n, i.e. an = Cn + D, where C and D are considered as constants, then the sequence is arithmetic progression.

Proof:

Let an be the last term

Then, Tn = a + (n – 1)d

an = a + nd – d

dn + (a-d)

an = Cn+D

Where C=d and D= a – d

• Property 5

A sequence is said to be an arithmetic progression if the sum of its first n terms is of the form Cn² + Dn, where C and D are constants independent of n. The common difference in such a case is 2A, i.e. 2 of the coefficient of n².

• Property 6

If the terms of an arithmetic progression are chosen at regular intervals, they form an AP.

• Property 7

If an, an + 1, and an + 2 are three consecutive terms of an arithmetic progression, then

2an + 1 =( an + an + 2)

So, these were the seven properties of arithmetic progression.

Properties of arithmetic

• Commutative Property- Addition and subtraction can be done with the help of this property, but multiplication and division cannot be done.

• Distributive Property: By a sum or difference, this property helps simplify the multiplication. Moreover, it simplifies mathematical calculations.

• Associative Property: This helps in rearranging the numbers without changing their value.

• Identity Element Property- When combined with other elements, other elements do not change.

Conclusion

In this article on the properties of arithmetic progression, first, we have discussed the meaning of arithmetic progression and arithmetic mean. Also, we have seen several properties of arithmetic progression with proofs to make the concept clear for the students. Appropriate formulas and properties of an arithmetic progression are essential to solve a question, as a minor error can make the whole question wrong. 

So, it is imperative to remember every property and concept of arithmetic progression to score decent marks in the exams, as questions related to arithmetic progression are not difficult but require a good IQ.