## Arithmetic Progression

### Introduction:

Consider the below in given list of numbers

- 2, 4, 6, 8,……….
- 140, 100, 60, 20, ……….
- -3, -2, -1, 0,……….

Each of the number in the above list is called a term.

In (i) each term is 2 more than the term preceding it

In (ii) each term is 40 less than the term preceding it

In (iii) each term is obtained by adding 1 to the term preceding it.

In the above list we observe that the successive terms are obtained by adding a fixed numbers to the preceding terms.

Such list of numbers forms an Arithmetic Progression.

### Definition:

An Arithmetic progression is defined as the list of numbers which are obtained by adding a fixed numbers to the preceding term (exceptional first term)

The fixed number which is added to the preceding term is called the common difference of the AP.

**Note:**

It is to be noticed that the common difference can be positive, negative or zero.

**Notation:**

We denote the first term of the AP by ‘a_{1}’, second term by ‘a_{2}’………….., n^{th }term by ‘a_{n}’ and the common difference by ‘d’ such that

a_{2}– a_{1}= a_{3}– a_{2}= a_{4}– a_{3…………………………………………………..= }a_{n}– a_{n-1}= d.

**Examples:**

(i) The weight (in kg) of some students of a school are.

20, 30, 40, …..80.

(ii) The number of unit squares in squares, with side 1, 2, 3,….. units, are respectively 1^{2}, 2^{2}, 3^{2 }_{………….}

(iii) The cash prize(in ₹) given by a School to the toppers of classes I to XII are respectively 200, 250, 300, 350, ……………., 750.

We see in the above example that each list of it form an AP.

Then we conclude that a, a+d, a+2d, a+3d,………… represents an Arithmetic Progression whose first term is ‘a’ and the common difference is ‘d’.

The above representation is known as the general form of an AP.

**Finite Ap:**

If tthe AP has only finite number of terms such an AP is called Finite AP. Also we see that finite AP has a last term.

**Infinite Ap:**

If the AP has infinite number of terms then such an AP is called an Infinite AP. Such an AP does not have a last term.

**Problem 1:**

Which of the following form an AP and write the next two terms

(i) 4, 10, 16, 22, …………………..

(ii) -2, 2, -2, 2, -2, …………………

**Solution:**

(i) We see that

a_{2}-a_{1 }= 10-4 = 6

a_{3}-a_{2} = 16-10 = 6

a_{4}-a_{3}= 22-16 = 6

i.e. a_{k+1}-a_{k }is the same all the time. So the above list of numbers form an AP with the common difference d= 6.

Thus the next two terms are 22+6= 28 and 28+6= 34.

(ii) We see that

a_{2}-a_{1}= 2- (-2) = 2+2 = 4

a_{3}-a_{2}= -2- (-2) = -2-2 = -4

Thus we observe that a_{3}-a_{2 }≠ a_{2}– a_{1.}

So we conclude that the above list of numbers does not form an AP.

**Problem 2:**

For the following AP’S write the first term and the common difference

(i) 3, 1, -1, -3, ……………………..

(ii) 0.6, 1.7, 2.8, 3.9, ………………….

**Solution:**

(i) We see that a_{1}= 3

(i.e) the first term of the AP is 3

and a_{2}– a_{1 }= 1-3= -2

a_{3}– a_{2}= -1- (1) = -1-1 = -2

a_{4}– a_{3}= -3- (-1) = -3+1= -2

i.e a_{k+1}– a_{k }is always the same

Hence the common difference d is -2.

(ii) We see that a_{1}= 0.6

(i.e) the first term of the AP IS 0.6

and a_{2- }a_{1 }= 1.7- 0.6= 1.1

a_{3}– a_{2}= 2.8- 1.7= 1.1

a_{4}– a_{3 }= 3.9- 2.8= 1.1

i.e. a_{k+1 }– a_{k }is always the same.

Hence the common difference d is 1.1.