# Sequence And Series

## Introduction to Sequences and Series

An itemized collection of elements in which repetitions of any sort is allowed is known as a sequence. They are very similar to sets and the members of the sequence are called elements (or terms). The length of a sequence is equal to the number of terms and it can be either finite or infinite. The primary difference between a sequence and a set is that in a sequence, individual terms can occur repeatedly in various positions.

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.

## Sequence and Series Definition

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, where 1,2,3,4,…..denotes the position of the term.

A sequence can be defined based upon 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 = $\sum_{n=1}^N$ a_n = a_1+a_2+a_3 + .. + a_N

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

## Sequence and Series Types

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 subtraction 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)

## 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 Ratio Successive term – Preceding term Example, a_2 – a_1 Successive term/Preceding term r = ar(n-1)/ar(n-2) General Term (nth Term) a_n = a + (n-1)d a_n = ar(n-1) nth term from the last term a_n = l – (n-1)d a_n = 1/r(n-1) Sum of first n terms s_n = n/2(2a + (n-1)d) s_n = a(rn-1)/(r-1) if r ≠ 1 s_n = na 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

Below are some common differences between sequences and 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 Questions

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.

c) 21st term as:  T21 = 4 + (21-1)3 = 4+60 = 64.

Question 2: Consider the sequence 1,4,16,64,256,1024….. Find: 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. Answer!

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#### Practise This Question

pth  term of the series (31n)+(32n)+(33n)+.... will be