Sequence and Series: Formula, Definition & Concepts

Sequence and Series: A sequence is a list of numbers arranged in a particular order, where each number in the list is called a term. Adding up all the terms of a sequence gives us a series. While sequences and series resemble sets, the key differences lie in sequences allowing repeated terms in different positions. The length of a sequence is determined by the number of terms it contains, which can be finite or infinite. Now lets begin exploring the formulas related to sequence and series.

Sequence and Series Formula

What are Sequence and Series?

A sequence is a structured arrangement of numbers, where each number is called a term. Typically, terms in a sequence are denoted as ai or an, with the index letter i or n showing their position. For instance, the second term of a sequence might be labeled a2, and a12 would represent the twelfth term.

A series refers to the total sum of all the terms within a sequence. However, for a series to exist, there must be a clear and consistent relationship among all the terms in the sequence.

S_N = a₁+a₂+a₃ + .. + a_n

Some Common Sequences

Arithmetic Sequences:

An arithmetic sequence is a series of numbers where each term is generated by adding or subtracting a fixed number from the previous term.

Geometric Sequences:

A geometric sequence is a series of numbers where each term is obtained by multiplying or dividing a fixed number with the preceding term.

Harmonic Sequences:

A sequence of numbers is considered a harmonic sequence if the reciprocals of all its elements form an arithmetic sequence.

Fibonacci Numbers:

The Fibonacci sequence is a series of numbers where each element is generated by adding the two preceding elements, starting with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

Sequence and Series Formulas

The nth term, an, of an Arithmetic Progression (A.P) where the first term is 'a' and the common difference is 'd' is given by the formula:

a_n=a+(n–1)d

 where, 

a= First Term

d= common difference

n=  position of term

1= last term

In conclusion, sequences and series provide a powerful tool for representing patterns, performing calculations, and analyzing various mathematical problems.

Also Read: Relations and Functions