Sequence and Series

Types of Sequences

In junior classes, you would have come across various patterns like,

 2, 4, 6,8, ……….

1, 3, 5, 7, ……….

5, 10, 15, 20 , ……..

and so on.

These patterns are generally known as sequences.

We can define a sequence as an arrangement of numbers in some definite order according to some rule.

We can commonly represent sequence as,

x1,,x2,x3,......xn. where 1,2, 3 are the position of the numbers and n is the nth term 

Whereas, series is defined as the sum of sequences.

Example: 1+2+3+4+.....+n, where n is the nth term 

Types of Sequences

Two main types of sequences are discussed here:

1. Arithmetic Sequence 

Consider the following sequence for instance,

1,4,7,10,13,16…….

Here, we can see that each term is obtained by adding 3 to the preceding term. This sequence is called Arithmetic sequence or Arithmetic Progression. It is also abbreviated as A.P. Thus we can define Arithmetic Sequence as

A sequence x1,,x2,x3,......xn. Is called an Arithmetic Progression, if there exists a constant number m such that,

x2 =  x1+ m 

x3 =  x2+ m 

x4=  x3+ m 

.

.

xn =  xn-1+ m and so on


The constant m is called the common difference of the A.P.

Thus we can write it as,

x, x+m, x+2m, x,+3m, ......x+(n-1)m


Where x is the first term

 m is a common difference.

2. Geometric Sequence

A sequence in which each term is obtained by either multiplying or dividing a certain constant number with the preceding one is said to be a geometric sequence. 

For example:

2,4,8,16,32,64,128...and so on

Here we can see that there is a common factor 2 between each term.

The geometric sequence can be commonly written as,


a, am, am2, am3…….

Where a is the first term

 m is the common factor between the terms.

Difference Between Sequence and Series

There is a bit of confusion between sequence and series, but you can easily differentiate between Sequence and series as follows:

  • A sequence is a particular format of elements in some definite order, whereas series is the sum of the elements of the sequence.

  • In sequence order of the elements are definite, but in series the order of elements is not fixed.

  • A sequence is represented as 1,2,3,4,....n, whereas the series is represented as 1+2+3+4+.....n.

  • In sequence, the order of elements has to be maintained, whereas in series the order of elements is not important.

Sequence and Series Formulas

Formulas for Arithmetic Sequence:

  • Sequence = x, x+m, x+2m, x,+3m, ......x+(n-1)m

            Where x is the first term

            m is a common difference.

  • Common difference = m = Successive term - Preceding term =  x2- x1

  • General term = nth term = x+(n-1)m

  • Sum of first nth terms = sn = n/2(2a + (n-1)d)

Formulas for Geometric Sequence

  • Sequence = a, am, am2, am3…….

Where a is the first term

            m is the common factor between the terms.

  • Common factor = m = Successive term / Preceding term 

             m = am(n-1)/am(n-2)

  • General term = nth term = an = am(n-1)

  • Sum of first n terms  =sn = a(mn)/(1-m) if m = 1

             sn = a(mn -1)/(m – 1)………(2) if m > 1

Formula for Series

sn= n/2( 1 + n)

Sequence and Series Examples

Example 1:

Write an A.P when its first term is 10 and the common difference is 3.

Solution:

Step 1: Arithmetic Progression = A.P. = a, a+m , a+2m , a +3m, a+4m.......

Step 2: here, a=10 and m = 3

So let put its value in the equation

Step 3: 10, 10+3, 10 +2*3, 10 + 3*3, 10 + 4 * 3 ………

We get,

10,13,16, 19, 22, …….

Example 2

Write the first three terms of the sequence defined as an = n2 + 1

Solution:

Step1: we have an = n2 + 1 


Step 2: Putting n = 1,2,3. We get


Step 3: a1 = 12 + 1 = 1 + 1 = 2

  a2 = 22 + 1 = 4 + 1 = 5

  a= 32 + 1 = 9 + 1 = 10


Step 4: Thus, the first three terms of the sequence an = n2 + 1 2, 5 and 10 respectively.

Quiz Time

  1. Write first five terms of each of the following sequence whose nth terms are:

1. an = 3n + 2

2.an ,= 3 n

  1. Write an A.P. having 4 as the first term and -3 as the common difference.

FAQ (Frequently Asked Questions)

1. How to Calculate Sequence and Series?

Ans: An arithmetic sequence is the of numbers in a particular order, such that the difference between the two terms is always constant

Suppose we have series 3, 9, 12, 15, 18……

If we find the difference between the two successive numbers it would be 3 and that remains constant.

And series is just the addition of the terms of an arithmetic sequence.

Suppose we have to find the sum of the arithmetic series 1,2,3,4 ...100

We have to just put the values in the formula for series

I.e. Sn= n/2 ( 1 + 100)

Put n = 100

S100 = 100/2 ( 1 + 100)

S100  = 50 * 101 = 5050

2. What is the nth Term Signified in Sequence?

Ans: If you don’t want to keep on adding the common difference or multiply the common factors to each term until you get the desired result. The nth term enables you to find any term of a sequence directly.  

Suppose you want to find the sum of 80 numbers you can directly substitute n = 80 in the given formula.

For example: To find the 25th term for the sequence 3, 6, 9, 12, 15,......

The formula for the sequence is an = n + (n- 1) d

Where n = 25 and d is difference = 3

We get a25 = 25 + 24*3

a25 =  25 + 72 = 97

So see how easy it is to calculate the nth term without actually estimating  each term.