Sequences And Series Formula, Definitions, Types, Concepts (2024)

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Sequences and Series Formula are concerning sequences and series are associated with various categories of mathematical sequences and series. A sequence comprises a collection of arranged elements that adhere to a particular pattern, while a series encompasses the summation of the elements within a sequence. The assortment of sequence and series formulas typically incorporates expressions for the nth term and the summation.

Sequences and series constitute fundamental topics within the realm of Arithmetic. A sequence denotes an enumerated assemblage of elements that permits repetition in any manner, while a series signifies the accumulation of all the elements. An instance commonly encountered in sequence and series is the arithmetic progression.

In succinct terms, a sequence embodies a roster of items or entities systematically arranged.

A series is a broader notion encapsulating the summation of all terms within a sequence. However, a distinct relationship among all sequence terms must be established.

What are Sequences and Series Formulas?

The following compilation encompasses formulas for arithmetic, geometric, and harmonic sequences and series. This collection comprises:

Formulas to compute the nth term within the sequence.
Formulas to ascertain the sum of the n terms in the series.

An arithmetic sequence involves a consistent difference between consecutive terms. A geometric sequence features a uniform ratio between successive terms. In contrast, a harmonic sequence exhibits an arithmetic sequence relationship among the reciprocals of its terms. The visual representation below displays all the formulas for sequences and series.

Now, let’s examine each of these formulas closely and gain a comprehensive understanding of the significance of each variable.

Also Check – Linear Equation Formula

Types of Sequence and Series

Some of the most common examples of sequences are:

Arithmetic Sequence and Series Formulas

Let’s ponder upon the arithmetic sequence denoted by a, a+d, a+2d, a+3d, a+4d, …. Here, ‘a’ signifies its initial term, and ‘d’ stands for its constant difference. Consequently:

The ‘nth’ term of this arithmetic sequence, represented as ‘an’, can be computed using the formula: an = a + (n – 1) d.

The total sum of the arithmetic series, denoted as ‘Sn’, can be calculated through the formula: Sn = n/2 (2a + (n – 1) d) (or) Sn = n/2 (a + an).

Also Check – Quadrilaterals Formula

Geometric Sequence and Series Formulas

Let’s examine the geometric sequence consisting of a, ar, ar², ar³, and so on, where ‘a’ is the initial term and ‘r’ signifies the common ratio. Thus:

The ‘nth’ term of this geometric sequence, indicated as ‘an’, can be calculated using the formula: an = a × r^(n – 1).

For the finite geometric series (the sum of the initial ‘n’ terms), denoted as ‘Sn’, the calculation is as follows: Sn = a × (1 – r^n) / (1 – r).

In the scenario of an infinite geometric series, denoted as ‘Sn’, the formula becomes: Sn = a / (1 – r) when |r| < 1. It’s important to note that the sum is undefined when |r| ≥ 1.

Also Check – Coordinate Geometry Formula

Harmonic Sequence and Series Formulas

Let’s explore the harmonic sequence which includes terms like 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), and so forth. Here, the starting term is ‘1/a’, and ‘d’ stands for the common difference of the arithmetic sequence a, a + d, a + 2d, and so on. Therefore:

To determine the ‘nth’ term of this harmonic sequence, denoted as ‘an’, we employ the formula: an = 1 / (a + (n – 1) d).

As for the summation of the harmonic series, represented as ‘Sn’, the calculation is: Sn = 1/d ln [ (2a + (2n – 1) d) / (2a – d) ].

Fibonacci Numbers

Fibonacci numbers constitute a captivating numerical sequence where each element is derived by summing the two preceding elements, and the sequence commences with the numbers 0 and 1. The sequence is defined as follows: F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2.