 # Sequence And Series

Sequence and series are the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series.

• In short, a sequence is a list of items/objects which have been arranged in a sequential way.
• A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

The fundamentals could be better understood by solving problems based on the formulas. They are very similar to sets but the primary difference is that in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms and it can be either finite or infinite.  This concept is explained in a detailed manner in Class 11 Maths. With the help of definition, formulas and examples we are going to discuss here the concepts of sequence as well as series.

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## Sequence and Series Definition

A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a1, a2, a3, a4,……… etc. denote the terms of a sequence, then 1,2,3,4,…..denotes the position of the term.

A sequence can be defined based on the number of terms i.e. either finite sequence or infinite sequence.

If a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by

SN = a1+a2+a3 + .. + aN

Note:  The series is finite or infinite depending if the sequence is finite or infinite.

## Types of Sequence and Series

Some of the most common examples of sequences are:

• Arithmetic Sequences
• Geometric Sequences
• Harmonic Sequences
• Fibonacci Numbers

### Arithmetic Sequences

A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.

### Geometric Sequences

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

### Harmonic Sequences

A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.

### Fibonacci Numbers

Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

Also, see:

## Sequence and Series Formulas

List of some basic formula of arithmetic progression and geometric progression are

 Arithmetic Progression Geometric Progression Sequence a, a+d, a+2d,……,a+(n-1)d,…. a, ar, ar2,….,ar(n-1),… Common Difference or Ratio Successive term – Preceding term Common difference = d = a2 – a1 Successive term/Preceding term Common ratio = r = ar(n-1)/ar(n-2) General Term (nth Term) an = a + (n-1)d an = ar(n-1) nth term from the last term an = l – (n-1)d an = l/r(n-1) Sum of first n terms sn = n/2(2a + (n-1)d) sn = a(1 – rn)/(1 – r) if |r| < 1 sn = a(rn -1)/(r – 1) if |r| > 1

*Here, a = first term, d = common difference, r = common ratio, n = position of term, l = last term

## Difference Between Sequences and Series

Let us find out how a sequence can be differentiated with series.

 Sequences Series Set of elements that follow a pattern Sum of elements of the sequence Order of elements is important Order of elements is not so important Finite sequence: 1,2,3,4,5 Finite series: 1+2+3+4+5 Infinite sequence: 1,2,3,4,…… Infinite Series: 1+2+3+4+……

## Sequence and Series Examples

Question 1: If 4,7,10,13,16,19,22……is a sequence, Find:

1. Common difference
2. nth term
3. 21st term

Solution: Given sequence is, 4,7,10,13,16,19,22……

a) The common difference = 7 – 4 = 3

b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d is the common difference.
Tn = 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1
c) 21st term as:  T21 = 4 + (21-1)3 = 4+60 = 64.

Question 2: Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term.

Solution: The common ratio (r)  = 4/1 = 4

The preceding term is multiplied by 4 to obtain the next term.

The nth term of the geometric sequence is denoted by the term Tn and is given by Tn = ar(n-1)
where a is the first term and r is the common ratio.

Here a = 1, r = 4 and n = 9

So, 9th term is can be calculated as T9 = 1* (4)(9-1)= 48 = 65536.

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Q1

### What does a Sequence and a Series Mean?

A sequence is defined as an arrangement of numbers in a particular order. On the other hand, a series is defined as the sum of the elements of a sequence.

Q2

### What are Some of the Common Types of Sequences?

A few popular sequences in maths are:

• Arithmetic Sequences
• Geometric Sequences
• Harmonic Sequences
• Fibonacci Numbers
Q3

### What are Finite and Infinite Sequences and Series?

Sequences: A finite sequence is a sequence that contains the last term such as a1, a2, a3, a4, a5, a6……an. On the other hand, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..

Series: In a finite series, a finite number of terms are written like a1 + a2 + a3 + a4 + a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a1 + a2 + a3 + a4 + a5 + a6 + ……an +…..

Q4

### Give an example of sequence and series.

An example of sequence: 2, 4, 6, 8, …
An example of a series: 2 + 4 + 6 + 8 + …

Q5

### What is the formula to find the common difference in an arithmetic sequence?

The formula to determine the common difference in an arithmetic sequence is:
Common difference = Successive term – Preceding term.

Q6

### How to represent the arithmetic sequence?

If “a” is the first term and “d” is the common difference of an arithmetic sequence, then it is represented by a, a+d, a+2d, a+3d, …

Q7

### How to represent the geometric sequence?

If “a” is the first term and “r” is the common ratio of a geometric sequence, then the geometric sequence is represented by a, ar, ar2, ar3, …., arn-1, ..

Q8

### How to represent arithmetic and geometric series?

The arithmetic series is represented by a + (a+d) + (a+2d) + (a+3d) + …
The geometric series is represented by a + ar + ar2 + ar3 + ….+ arn-1+ ..

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7. End term, a_n = 3n(2ⁿ-1)
a_1 = 3(1)(2^1-1) = 3
a_2 = 3(2)(2^2-1) = 18
a_3 = 3(3)(2^3-1) = 63
a_4 = 3(4)(2^4-1) = 180
Therefore, the first four terms of sequence are 3,18,63 and 180.

• The finite sequence has first term and last term, thus it has an end.
For example, set of numbers from 1 to 10 is a finite sequence.
Infinite sequence does not have last term, it keeps on going infinitely.
For example, a set of natural numbers has no end.

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