Maths Sequence And Series

Sequence And Series

Maths Sequence And Series
Written by Mary Christina

Arithmetic Progression

Introduction:

Consider the below in given list of numbers

  1. 2, 4, 6, 8,……….
  2. 140, 100, 60, 20, ……….
  3. -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 ‘a1’, second term by ‘a2’………….., nth term by ‘an’ and the common difference by ‘d’ such that

a2– a1= a3– a2= a4– a3…………………………………………………..= an– an-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  12, 22, 32 ………….

Sequence and series square

(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

a2-a1 = 10-4 = 6

a3-a2 = 16-10 = 6

a4-a3= 22-16 = 6

i.e. ak+1-ak 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

a2-a1= 2- (-2) = 2+2 =  4

a3-a2= -2- (-2) = -2-2 = -4

Thus we observe that a3-a2 ≠ a2– a1.

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 a1= 3

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

and  a2– a1 = 1-3= -2

a3– a2= -1- (1) = -1-1 = -2

a4– a3= -3- (-1) = -3+1= -2

i.e ak+1– ak is always the same

Hence the common difference d is -2.

(ii) We see that a1= 0.6

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

and  a2- a1 = 1.7- 0.6= 1.1

a3– a2= 2.8- 1.7= 1.1

a4– a3 = 3.9- 2.8= 1.1

i.e. ak+1 – ak is always the same.

Hence the common difference d is 1.1.

About the author

Mary Christina

Mary Christina from Chennai, India. She has completed Master Degree in Mathematics and she loves to teach and share about her knowledge in maths for High School and College Students...