## Arithmetic Sequence

 First Number Common Difference nth Number to obtain

## Geometric Sequence

 First Number Common Ratio nth Number to obtain

## Fibonacci Series

 Nth number to obtain

# SEQUENCE CALCULATOR

Numbers are said to be in sequence if they follow a particular pattern or order. Let us discuss three types of sequences in this post: Arithmetic, Geometric and Fibonacci.

### Arithmetic Sequence

The terms a1, a2, a3, a4, a5, ……an are said to be in an arithmetic progression P, when a2-a1=a3-a2, i.e. when the terms increase or decrease continuously by a common value. This common value is called the common difference (d) of the Arithmetic Sequence.

The first term of the P is denoted as ‘a’ and the number of terms is denoted as ‘n’. We also call an arithmetic sequence as an arithmetic progression.

Common Difference is the difference between the successive term and its preceding term. It is always constant for the arithmetic sequence.

Common difference (d) = a2 – a1

To find the nth term of an arithmetic sequence, we use

Tn = a + ( n - 1 ) × d

Sum of terms of an Arithmetic sequence is

 Sn
=
 [ n × ( 2a + ( n - 1 ) × d ) ] 2

Where ‘a’ is the first term and ‘d’ is the common difference.

Examples of Arithmetic Progression:

1. 5, 7, 9, 11, 13, ...

Here ‘5’ is the first term and the common difference d = 7-5 = 2

2. 15, 12, 9, 6, 3, 0, -3, -6 ,…

Here ‘15’ is the first term and the common difference d = 12-15 = -3

### Geometric Sequence

The terms a1, a2, a3, a4, a5……an are said to be in geometric sequence, when a2/a1=a3/a2=r, where ‘r’ is called the common ratio(r) of the Geometric Sequence.

The first term of the sequence is denoted as ‘a’ and the number of terms is denoted as ‘n’. We also call a geometric sequence as a geometric progression.

Common Ratio is the ratio between the successive term and its preceding term. It is always constant for a given geometric sequence.

 Common Ratio(r)
=
 a2 a1

To find the nth term of a geometric sequence, we use

Tn = a × r ( n - 1 )

Sum of terms of a geometric sequence is

 Sn
=
 a ( rn - 1 ) ( r - 1 )

where ‘a’ is the first term and ‘r’ is the common ratio.

Sum of terms of an infinitely long decreasing geometric sequence is

 Sn
=
 a ( 1 - r )

Examples of Geometric Progression:

1. 1, 2, 4, 8, 16….

Here ‘1’ is the first term and the common ratio(r) = 2/1 = 2

2. -1, 2, -4, -8, -16…

Here ‘-1’ is the first term and the common ratio(r) = 2/(-1) = -2

### Fibonacci Series

The Fibonacci Sequence is the series of numbers - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number of the series is obtained by adding up the two numbers before it.

• The number 2 is obtained by adding the 1 and before it. (1+1)
• The number 3 is obtained by adding the 2 and 1 before it (1+2),
• And number 5 is (2+3), and so on!

### Do you know these facts?

• Leonardo Pisano Bogollo also was known as "Fibonacci", and lived between 1170 and 1250 in Italy. "Fibonacci" means "Son of Bonacci".
• Fibonacci Day: November 23rd is called a Fibonacci day, as it has the digits "1, 1, 2, 3" which is a part of the sequence.

### How to use CalculatorHut’s sequence calculator?

You can use our series calculator for calculating and finding out any numbers from arithmetic, geometric and Fibonacci series.