## HOW TO KNOW IF THE GIVEN SEQUENCE IS AN ARITHMETIC PROGRESSION

How to know if the given sequence is an arithmetic progression :

Only if the common difference are equal the given sequence is called an arithmetic progression.

To find the common difference, we need to subtract the second term from the first term. The answer must be equal to the difference of any term with the previous term.

That is,

a3 - a2  =  a4 - a3  =  d (a2- a1)

Let us look into some example problems to understand the above concept.

Example 1 :

Determine whether the given sequence is an arithmetic sequence. If it is, state the common difference.

7, 6, 5, 4, …

Solution :

To check if the given sequence is an arithmetic sequence, let us find the common difference.

 d  =  a2 - a1  =  6 - 7   =  -1 d  =  a3 - a2  =  5 - 6   =  -1 d  =  a4 - a3  =  4 - 5   =  -1

Since the common difference are same, the above sequence is arithmetic progression.

Let us see the next example on "How to know if the given sequence is an arithmetic progression".

Example 2 :

Determine whether the given sequence is an arithmetic sequence. If it is, state the common difference.

10, 12, 15, 18, …

Solution :

To check if the given sequence is an arithmetic sequence, let us find the common difference.

 d  =  a2 - a1  =  12 - 10   =  2 d  =  a3 - a2  =  15 - 12   =  3 d  =  a4 - a3  =  18 - 15   =  3

Since the common difference are not same, the above sequence is not arithmetic progression.

Let us see the next example on "How to know if the given sequence is an arithmetic progression".

Example 3 :

Determine whether the given sequence is an arithmetic sequence. If it is, state the common difference.

9, 5, -1, -5, …

Solution :

To check if the given sequence is an arithmetic sequence, let us find the common difference.

 d  =  a2 - a1  =  5 - 9   =  -4 d  =  a3 - a2  =  -1 - 5   =  -6 d  =  a4 - a3  =  -5 - (-1)   =  -5 + 1=  -4

Since the common difference are not same, the above sequence is not arithmetic progression.

Example 4 :

Determine whether the given sequence is an arithmetic sequence. If it is, state the common difference.

-15, -11, -7, -3, …

Solution :

To check if the given sequence is an arithmetic sequence, let us find the common difference.

 d  =  a2 - a1  =  -11 - (-15)   =  -11 + 15=  4 d  =  a3 - a2  =  -7 - (-11)   =  -7 + 11=  4 d  =  a4 - a3  =  -3 - (-7)   =  -3 + 7=  4

Since the common difference are same, the above sequence is arithmetic progression.

Example 5 :

Determine whether the given sequence is an arithmetic sequence. If it is, state the common difference.

-0.3, 0.2, 0.7, 1.2, …

Solution :

To check if the given sequence is an arithmetic sequence, let us find the common difference.

 d  =  a2 - a1  =  0.2 - (-0.3)   =  0.2 + 0.3=  0.5 d  =  a3 - a2  =  0.7 - 0.2  =  0.5 d  =  a4 - a3  =  1.2 - 0.7  =  0.5

Since the common difference are same, the above sequence is arithmetic progression.

Example 6 :

Determine whether the given sequence is an arithmetic sequence. If it is, state the common difference.

2.1, 4.2, 8.4, 17.6, …

Solution :

To check if the given sequence is an arithmetic sequence, let us find the common difference.

 d  =  a2 - a1  =  4.2 - 2.1   =  2.1 d  =  a3 - a2  =  8.4 - 4.2  =  4.2 d  =  a4 - a3  =  17.6 - 8.4  =  9.2

Since the common difference are not same, the above sequence is not arithmetic progression.

After having gone through the stuff given above, we hope that the students would have understood "How to know if the given sequence is an arithmetic progression".

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