Arithmetic Progressions

Any given series of numbers may exhibit some special properties. A sequence of numbers in arithmetic progression is one of these special series. Look at the given video to understand the concept of arithmetic progression.

Let us look at the following example.

**Example 1:**

**Check whether the following sequences form an A.P. or not.**

**(i) **

**(ii) **

**(iii) **

**(iv) A list of prime numbers greater than 2**

**(v) A list of the squares of natural numbers**

**Solution:**

**(i)**The given sequence is

Difference between the second and the first term

Difference between the third and the second term

Difference between the fourth and the third term

Since the difference between any two consecutive terms is a constant, the sequence is an arithmetic progression.

**(ii)**The given sequence is

Difference between the second and the first term

Difference between the third and the second term

Difference between the fourth and the third term

Since the difference between any two consecutive terms is a constant, the sequence is an arithmetic progression.

**(iii)**The given sequence is

This sequence can be rewritten as

Difference between the second and the first term

Difference between the third and the second term

Difference between the fourth and the third term

Since the difference between the consecutive terms is not a constant, the sequence is **not** an arithmetic progression.

**(iv)**The list of the prime numbers greater than 2 is: 3, 5, 7, 11 …

Difference between the second and the first term = 5 – 3 = 2

Difference between the third and the second term = 7 – 5 = 2

Difference between the fourth and the third term = 11 – 7 = 4

Since the difference between the consecutive terms is not a constant, the sequence is **not** an arithmetic progression.

**(v)**The list of the squares of natural numbers is: 1^{2}, 2^{2}, 3^{2}, 4^{2} … = 1, 4, 9, 16 …

Difference between the second and the first term = 4 – 1 = 3

Difference between the third and the second term = 9 – 4 = 5

Difference between the fourth and the third term = 16 – 9 = 7

Since the difference between the consecutive terms is not a constant, the sequence is **not** an arithmetic progression.

**An arithmetic progression (A.P.) is a sequence in which the difference between any two consecutive terms is constant.**

Now there are many common terms associated with an arithmetic progression, which one has to understand to solve any A.P. related problem. So, let us look at the given video to understand the meaning of the terminology related to a...

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