Definition of Sequence and Series

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Sequence: Sequence and Series is one of the most important concepts in Arithmetic. A sequence refers to the collection of elements that can be repeated in any sort. In other words, we can say a sequence refers to the list of objects or items that have been arranged in a systematic and sequential manner.

Eg: a1,a2,a3, a4…….

Series: Whereas, Series refers to the sum of all the elements available or we can say A series can be referred to as the sum of all the elements available in the sequence. One of the most common examples of a sequence and series would be Arithmetic Progression.

Eg: If

a1,a2,a3, a4…….   etc is considered to be a sequence, then the sum of terms in the sequence a1+a2+a3+ a4……. are considered to be a series. 

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,

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,

Where a is the first term

m is the common factor between the terms.

A few other types of Sequence and Series include

3. Harmonic Sequences

Harmonic Sequences refer to a series of numbers that are said to be in a harmonic sequence. These series of numbers are said to be in a harmonic sequence only if the reciprocal of all the elements that are a part of the sequence are created into an arithmetic sequence. 

4. Fibonacci Number Sequence

Fibonacci Numbers are a form of a number sequence where every element can be obtained by adding two elements. With this, the sequence starts with 0 and 1. Hence, the Fibonacci Number Sequence is defined as 

F1=0, 

F2 = 1, 

F3 = 1,

F4 = 2,

F5 = 3,

…

..

Fn = Fn-1 + Fn-2

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 a 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 = \[\frac{n}{2}(2a + (n-1)d)\]

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 = \[\frac{am^{(n-1)}}{am^{(n-2)}}\]

• General term = nth term = an = \[am^{(n-1)}\]

• Sum of n terms in GP = Sn = na if m = 1,

• Sum of n terms in GP = Sn = \[\frac{a(m^{n}-1)}{(m – 1)}\] when m > 1

• Sum of n terms in GP = Sn = \[\frac {a(1- m^{n})}{(1 – m)}\] when m < 1

Formula for Series

\[s_{n}=\frac{n}{2( 1 + n)}\]

Sequence and Series Solved 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

  a3  = 32 + 1 = 9 + 1 = 10

Quiz Time

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

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

3. Write a G.P. having 4 as the first term and 2 as the common ratio. 

Summary Points

• In a math series and sequence, a is considered to be the first term, d is the common difference, and an is known to be the nth term. 

• The arithmetic sequence can be explained as a, a + d, a + 2d, a + 3d, …

• Each term in a geometric progression is obtained by multiplying the common ratio of the successive term to the preceding term.

• The geometric progression formula is known as :

an = arn-1 

Sn = a/(1-r) Where |r| < 1. 

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