**Sequences and series Grade 12**

Table of Contents

**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.

## 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 a_{1}, a_{2}, a_{3}, a_{4},……… 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 a_{1}, a_{2}, a_{3}, a_{4,} ……. is a sequence, then the corresponding series is given by

S_{N} = a_{1}+a_{2}+a_{3} + .. + a_{N}

**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, F_{0} = 0 and F_{1} = 1 and F_{n} = F_{n-1} + F_{n-2}

**Chapter 1: Sequences and series**

**1.1 Arithmetic sequences****1.2 Geometric sequences****1.3 Series****1.4 Finite arithmetic series****1.5 Finite geometric series****1.6 Infinite series****1.7 Summary****End of chapter exercises****Practice this chapter**

## 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**

**Sequence and Series Examples**

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

**Common difference****nth term****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 T _{n} and is given by T_{n} = a + (n-1)d, where “a” is the first term and d is the common difference.**

**T**_{n}= 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1

**c) 21st term as: T**_{21}= 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 T _{n} and is given by T_{n} = 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 T _{9} = 1* (4)^{(9-1)}= 4^{8} = 65536.**

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**Frequently Asked Questions**

**Frequently Asked Questions**

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

**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.**

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

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

**A few popular sequences in maths are:**

**Arithmetic Sequences****Geometric Sequences****Harmonic Sequences****Fibonacci Numbers**

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

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

**Sequences: A finite sequence is a sequence that contains the last term such as a _{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}……a_{n. }On the other hand, an infinite sequence is never-ending i.e. a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}……a_{n….}.**

**Series: In a finite series, a finite number of terms are written like a _{1 }+ a_{2 }+ a_{3} + a_{4 }+ a_{5} + a_{6 + }……a_{n}. In case of an infinite series, the number of elements are not finite i.e. a_{1 }+ a_{2 }+ a_{3} + a_{4 }+ a_{5} + a_{6 + }……a_{n }+_{…..}**

**Give an example of sequence and series.**

**Give an example of sequence and series.**

**An example of sequence: 2, 4, 6, 8, …**

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

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

**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.**

**How to represent the arithmetic sequence?**

**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, …**

**How to represent the geometric sequence?**

**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, ar ^{2}, ar^{3}, …., ar^{n-1}, ..**

**How to represent arithmetic and geometric series?**

**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 + ar ^{2} + ar^{3} + ….+ ar^{n-1}+ ..**