PROPERTIES OF ARITHMETIC PROGRESSION

In this section, you will learn the properties of arithmetic progression. 

Properties of Arithmetic Progression

Property 1 : 

In an arithmetic progression, if every term is added by a constant, then the resulting sequence will also be an arithmetic progression. 

Example : 

Consider the arithmetic progression given below. 

3, 6, 9, 12,........................... 

Add 2 to every term in the above sequence.

5, 8, 11, 14,...........................

The above sequence is an arithmetic progression. 

Property 2 : 

In an arithmetic progression, if every term is subtracted by a constant, then the resulting sequence will also be an arithmetic progression. 

Example : 

Consider the arithmetic progression given below. 

3, 6, 9, 12,........................... 

Subtract 1 from every term in the above sequence.

2, 5, 8, 11,...........................

The above sequence is an arithmetic progression. 

Property 3 : 

If every term is multiplied by a non-zero number, then the resulting sequence will also be an arithmetic progression.

Example : 

Consider the arithmetic progression given below. 

3, 6, 9, 12,........................... 

Multiply every term in the above sequence by 2.

6, 12, 18, 24,...........................

The above sequence is an arithmetic progression. 

Property 4 : 

If every term is divided by a non-zero number, then the resulting sequence will also be an arithmetic progression.

Example : 

Consider the arithmetic progression given below. 

3, 6, 9, 12,........................... 

Divide every term in the above sequence by 3.

1, 2, 3, 4,...........................

The above sequence is an arithmetic progression. 

Property 5 : 

If the three terms (a, b, c) are in arithmetic progression, then two times of the middle term is equal to sum of the first term and last term.

That is, 

2b  =  a + c

or

If the three terms (a, b, c) are in arithmetic progression, then the middle term is equal to average of the first term and last term.

That is, 

b  =  (a + c) / 2

Property 6 : 

A sequence can be considered as arithmetic progression, if and only if its n^th term (a_n) is expressed as a linear equation in the form 

a_n  =  An + B

where A and B are constants.

Here, the common difference of the arithmetic progression is A. That is, the coefficient of 'n'.  

Property 7 : 

A sequence can be considered as arithmetic progression, if and only if the sum of its first n terms (S_n) is in the form

S_n  =  An² + Bn

where A and B are constants.

Here, the common difference of the arithmetic progression is 2A. That is, two times the coefficient of n².  

Property 8 : 

In an arithmetic progression, the terms which are selected at a regular interval will form an arithmetic progression. 

Example :

Consider the arithmetic progression given below. 

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33........................... 

The terms in red color in the above arithmetic progression are in regular intervals.

Form a sequence with the terms in red color. 

3, 9, 21, 30, ...........................

The above sequence is an arithmetic progression. 

Property 9 : 

In an arithmetic progression with finite number of terms, the sum of two terms which are equidistant from the beginning and end will always be equal to sum of the first term and last term  of the same arithmetic progression.

Example :

Consider the arithmetic progression given below. 

3, 6, 9, 12, 15, 18........................126, 129, 132, 135, 138, 141 

Find the sum of two terms 9 and 135 which are equidistant from the beginning and end. 

9 + 135  =  144 -----(1)

Find the sum of the first term 3 and last term 141. 

3 + 141  =  144 -----(2)

From (1) and(2), the above said property is true.  

Property 10 : 

In an arithmetic progression having a common difference of zero is called a constant arithmetic progression.

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