Sequence and Series in detail

Understanding About Sequence And Series

The concept of sequence and series can be well understood by reading more about it in detail. In Mathematics, a sequence is a list of elements appearing in disciplined order. Just like a set, a sequence consists of elements, terms, or members. The number of terms present in the list/orderly manner depicts the length of the sequence wherein the length of a sequence can be infinite. 

Unlike a set, the order in which the elements are present in a sequence matters.

A sequence is the sequential arrangement of numbers. Sequence and series are commonly used in mathematics. A sequence is also known as a progression from which a series is and can be developed. One of the basic concepts in the field of Algebra can be said to be sequence and series. 

A sequence is the grouped arrangement of numbers in an orderly manner and according to some specific rules, whereas on the other side a series is the sum of the elements in the sequence.

In another word a sequence can be defined as a function that has a countable domain in an orderly set, for example- a set of natural numbers, set of even numbers, set of odd numbers, etc.

For example, 1, 3, 5, 7 is a sequence with four elements and the corresponding series will be 1 + 3 + 5 + 7 where the sum of the series or value of the series will be 16.

Sequences can be of two types-

Infinite Terms Sequence  and  Finite Terms Sequence

A sequence is finite if it has a countable number of terms or terms that end and infinite if it does not have an end, that is the sequence is never-ending.

Finite sequence: {4,8,12,16,…,64}

Infinite sequence: {4,8,12,16,20,24,…∞}

A sequence is a set of numbers or sequences in a given order or set of rules. An array is formed by adding the digits of a sequence. Sequences can be of two types, that is, an infinite series and finite series depending on the order and the terms of the sequence. An infinite number of words in the array is also possible in some cases.

Let’s understand it with an example. 2, 4, 6, 8, 10, 12, … is a sequence with a common difference of 2 between any two elements and the sequence continues to infinity until the upper limit is given. Such sequences are called arithmetic arrays. Now if we add numbers in a row like 2 + 4 + 6 + 8 + 10 … it forms the required array. 

Examples of Sequences and Series-

Arithmetic Series- 200 + 400 + 600 + 800 + 1000 + …..+ n.

Arithmetic Sequence- 200, 400, 600, 800, 1000, …., n.

Geometric Series- 64 + 32 + 16 + 8 + 4 + 2 + …..+ n.

Geometric Sequence- 64, 32, 16, 8, 4, 2, ……, n.

Types of Sequences 

There are various types of sequences and series. Below mentioned are the most commonly used sequences. The types of sequence and series are:

• Arithmetic Sequences (Arithmetic progression)

• Geometric Sequences (Geometric progression)

• Harmonic Sequences (Harmonic progression)

Arithmetic Sequences (Arithmetic progression)

An arithmetic progression (AP) is a number sequence where each term of the given sequence is thus found by adding a fixed number/difference to the previous term. This fixed number which is added is called the common difference.

AP (Arithmetic Progression) 

1,3,5,7,9,………∞

The successive terms can be found by adding 2 to the previous term

1+2= 3

3+2= 5

5+2= 7

7+2= 9

So, common difference for this AP is 2. 

Geometric Sequences  (Geometric progression)

Geometric Sequence is a type of progression in which to get the next term in the geometric progression, we will have to multiply each element of the sequence with a fixed term known as the common ratio, every time.

GP (Geometric progression) 

3,9,27,81,243,….. ∞

The secession terms can be found by multiplying 3 to the previous term. 

3×3=9

9×3=27

27×3=81

81×3=243

Harmonic Sequences (Harmonic progression)

Harmonic progression of a sequence of numbers is that if given terms or elements are in Arithmetic Progression then the reciprocal of the terms are said to be in Harmonic Progression. In simple terms, p, q, r, s, t, u are considered to be in Arithmetic Progression then  1p, 1q,1r,1s,1t,1u are in a Harmonic Series.

HP(Harmonic Progression)

⅕, 1/10, 1/15, 1/20,……….. ∞

Reciprocal of the following fractions are in AP with a common difference of 5. 

Difference Between Sequence and Series

The important differences between sequence and series are explained in the table given below:

Formulae for Sequences and series

There are various formulas related to various sequences and series by using them we can find a set of unknown values like the first term, nth term, common differences, etc. These formulas are different for each kind of sequence and series.

Series and Sigma Notation

Sigma notation, denoted by the uppercase Greek letter sigma (Σ), is used to represent summations, which is a series of numbers to be added together. Summation is the term given for the addition of a sequence of numbers, resulting in a sum.

Formulae for Arithmetic Progression

General Representation of Arithmetic sequence- 

a, a +d, a + 2d, a + 3d, … 

First term- a

Common difference(d) = Successive term – Preceding term 

nth term an = a + (n-1)d

Sum of arithmetic series Sn = (n2) (2a + (n-1)d) 

Formulae for Geometric Progression

General Representation of Geometric sequence– a, ar, ar²…., 

Geometric series- a, ar, ar²+…. 

First term- a

Common ratio- r

nth term – ar(n-1)

Sum of geometric series-

Finite series:

Sn =a(1 – r)1 – r when r < 1,

Sn = a x n when, r = 1

Infinite series: 

Sn = a(r – 1)r – 1 when r > 1.

Formulae for Harmonic Progression

1. If a, b, c, d… is a given Arithmetic Progression then if 1a,1b,1c,1d, is a Harmonic Progression.

 2. If two non-zero numbers are a and b, then the harmonic mean of a and b is a number H such that the sequence a, H, b is a H.P. We have.

1H =12(1a+1b) or H=2aba + b If a₁, a2… An are n numbers which are non-zero, then a+b harmonic mean H of these numbers is given by 

1H = 1n(1a1+1a2+….+1gAx)AH = G² when A > G >H.

If nth term of a series is Tn = an²+ bn c, then the sum of its nth terms is given by Sn = a Σ of n² +b Σ n + cn. In general Sn = ∆ Tn.

Conclusion 

The concept of sequence and series has been explained in much detail. The guide gives a comprehensive explanation of the topic as well as the elements associated with it.