## Series

In mathematics, a sequence is referred to as a systematic list of numbers. The numbers in the list are actually the terms of the sequence. A series is the summation of all the terms of a sequence. Sequence and series are like sets. However, the only difference between them is that in a sequence, individual terms can take place repeatedly in different positions. The length of a sequence is equivalent to the number of terms, which could either be finite or infinite. Below we will also learn the number series formula with which we are able to solve the number series math questions.

### Number Series Formula

The sum of the first n terms of a sequence is what we call a series.

If a sequence is geometric or arithmetic then there are specified formulas to calculate the sum of the first n terms, represented as S\[_{n}\], without really adding all of the terms.

### Formula for Sum of the Terms of an Arithmetic Series

In order to calculate the sum of the first n terms of an arithmetic sequence, we use the following formula,

S\[_{n}\] = n (a\[_{1}\] + a\[_{2}\])/2

Where,

n = number of terms

a\[_{1}\] = the first term

a\[_{n}\] = the last term.

Note: In mathematical terms, a sequence can neither be arithmetic nor geometric.

### Formula for Sum of the Terms of a Geometric Series

In order to calculate the sum of the first n terms of a geometric sequence, we use the following formula,

S\[_{n}\] = a\[_{1}\] (1−r\[^{n}\])/1−r, r≠1

Where,

n = number of terms

a\[_{1}\] = the first term

r = common ratio

### Sequence and Series

A sequence is referred to as an ordered list of numbers. The numbers in that list are known as the terms of the sequence. The terms of a sequence are generally named as a\[_{i}\] or a\[_{n}\], having a subscripted letter i or n being the index. Thus, the second term of a sequence might be named a\[_{2}\], and a\[_{12}\] will be the twelfth term.

A series on the other hand is called the sum of all the terms in a sequence. But, there has to be a definite link between all the terms of the sequence.

S\[_{N}\] = a\[_{1}\] + a\[_{2}\] + a\[_{3}\] +... + a\[_{n}\]

### Solved Examples

Example:

Determine the sum of the first 50 terms of the arithmetic sequence

Solution:

2, 5, 8, 11, 14, 19, 22, 25, 28, 31⋯

We will have to first find the 50th term:

A\[_{50}\] = a1+ (n−1) d

=2+49(3)

=149

Then find the sum:

S\[_{n}\] = n(a\[_{1}\] + a\[_{n}\])/2

S\[_{50}\] = 50(2 + 149)/2

=3775

Example:

Identify S\[_{10}\] of the geometric series 24+12+6+......

Solution:

First, we will find r

r = r\[_{2}\]/r\[_{1}\] = 12/24

=12

Now, we will find the sum:

S\[_{10}\] = 24(1− (1/2)\[^{10}\])/1−1/2

=3069/64

Q1. What are the Different Types of Sequence and Series?

Answer: Sequences: A finite sequence pause at the termination of the list of numbers like a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8},……a_{n}, whereas, an infinite sequence is never-ending i.e. it keeps continuing
a1, a2, a3, a4, a5, a6, a7, a8, a_{9}, a_{10}, a_{11}, a_{12},……a_{n} ……a_{n}

Series: In a finite series, a finite number of terms are expressed as
a1, a2, a3, a4, a5, a6, a7, a8,……a_{n}. Whereas, in the case of an infinite series, the number of elements is not finite i.e.
a1, a2, a3, a4, a5, a6, a7, a8, a9, a10,……a_{n}……a_{n}

Other common types of sequences include Arithmetic Sequences, Geometric Sequences, and Harmonic Sequences.

Q2. What are Fibonacci Numbers?

Answer: Fibonacci numbers are the numbers of the list that forms a sequence of numbers in which every element is acquired by adding two preceding elements and the sequence begins with 0 and 1. Sequence is described as, F_{0} = 0 and F_{1} = 1 and F_{n} = F_{n-1} + F_{n-2}