Sequences and Series

Introduction

One of the important concepts of Arithmetic is sequence and series. A series is the sum of the elements in a sequence, whereas a sequence is the grouped arrangement of numbers methodically and according to specified principles. For example, if a four element sequence is 1, 3, 5, and 9, the corresponding series will be 1 + 3 + 5 + 9, with the sum or value of the series being 18.

Sequences

The length of the sequence is defined as the number of ordered elements (potentially infinite). Unlike a set, order matters, and a phrase might appear often in the sequence at different points.

For example, (F, I, L, E) is a letter sequence that varies from (L, I, F, E) because the order matters, and (1, 1, 2, 3, 5, 8) is a legitimate sequence since the number 1 appears twice. Sequences can be finite, as in this case, or endless, as in the case of all even positive integers (2, 4, 6). Finite sequences are sometimes referred to as strings or words, whereas infinite sequences are called streams.

It is occasionally possible to find a formula for the sequence’s general term given multiple terms in the series. When a value for the integer n is entered into the formula, it will yield the nth term.

If a polynomial creates a sequence, it can be determined by observing whether the computed differences become constant over time.

•       Linear polynomials :

For example

  S = 3, 9, 15, 21,……

The starting term is 3 and the common number gap is 6. Hence the nth number of the sequence is 3 + (n – 1) * 6.

•     Quadratic polynomials :

Quadratic sequences of numbers are distinguished because the difference between terms changes by the same amount every time.

For example,

2, 7, 14, 23, 34, 47

We can see that the second difference is continuous and not equal to zero, indicating that the sequence is quadratic.

The sequence is a collection of numbers arranged in a specific order or according to a set of criteria. Sequences can be classified as  infinite terms sequence and finite terms sequence

Series

The digits from a sequence are added together to form a series.

The term ‘infinite series’ indicates that a series might have unlimited terms. Because the Riemann series theorem asserts that a so-called conditionally convergent series can be converged to any desired value or diverge by rearrangement of terms, the order of the terms in a series might be important.

The sum of the terms in a sequence is called a series. The nth partial sum is denoted by the sum of the first n terms.

Types of sequence and series

There are mainly three types of sequence and series:

1.     Arithmetic sequence and series:

An arithmetic sequence is when each term is either the addition or subtraction of a common term known as the common difference. An arithmetic sequence is, for example, 1, 3, 5, 9,… The arithmetic series is a series formed by using an arithmetic sequence. For example, 1 + 3 + 5 + 9… is an arithmetic series.

Arithmetic sequence formula: a, a + d, a + 2d, a + 3d…

Arithmetic series formula:  a + (a + d) + (a + 2d) + (a + 3d) + …

2.     Geometric sequence and series:

A geometric sequence is one in which all the terms have the same ratio. An arithmetic sequence is, for example, 2, 8, 32, 128,… A geometric series is a series formed by using geometric sequences. For example, 2 + 8 + 32 + 128… is a geometric series.

Geometric sequence formula : a, ar, ar2,…., ar(n-1),…

Geometric series formula:  a + ar + ar2 + …+ ar(n-1) + …

 There are two types of geometric progression:

•    finite geometric  progression
•    infinite geometric series.

3.     Harmonic sequence and series:

A harmonic sequence is one in which each term of an arithmetic sequence is multiplied by the reciprocal of that term. A harmonic series is 1, 1/4, 1/7, 1/10, and so on. The harmonic series is a series formed by using harmonic sequences. For example, 1 + 1/4 + 1/7 + 1/10…. is a harmonic series.

Conclusion

Sequences and Series play a significant part in many facets of our lives. They help us in decision-making by predicting, evaluating, and monitoring the consequences of a situation or an event. Various formulas result in many mathematical sequences and series. In calculus, physics, analytical functions, and many other mathematical tools, series such as the harmonic series and alternating series are extremely useful. They are commonly used in computer science, finance, and economics among other fields, to find numerous options for a given circumstance or set of criteria to design, analyse or predict.