# Sequence Calculator

Use this number sequence calculator to easily calculate the n-th term of an arithmetic, geometric or fibonacci sequence, and the sum of all terms between the starting number and the nth term.

## What is a sequence?

In mathematics, a **sequence** is an **ordered list of objects**, usually numbers, in which repetition is allowed. The number of elements is the **length** of the sequence. Number sequences can be expressed as the function that generates the next term in a sequence from the previous one.

Sequences can be **monotonically increasing** - that is if each term is greater than or equal to its preceding term, or they can be **monotonically decreasing**, if the reverse is true. If each element is larger than or smaller than the preceding element, then a sequence is strictly monotonically increasing or strictly monotonically decreasing, respectively. Our sequence calculator outputs **subsequences** of the specified sequence around the selected nth element.

## Using this sequence calculator

This is a very versatile calculator that will output sequences and allow you to calculate the sum of a number sequence between a starting item and an n-th term, as well as tell you the value of the n-th term of interest.

Start by selecting the type of sequence: you can choose from the arithmetic sequence (addition), geometric sequence (multiplication), and the special Fibonacci sequence. Then specify the direction of the sequence: increasing or decreasing, and the number you want to start from. Specify the common difference, which is how the sequence is constructed basically. This is the ratio between the elements. Finally, input which term you want to obtain using our sequence calculator. The fibonacci sequence is fixed as starting with 1 and the difference is prespecified.

The calculator output is a part of the sequence around your number of interest and the sum of all numbers between the starting number and the nth term of the sequence. See our sigma notation calculator for summing up series defined by a custom expression.

## Arithmetic Sequence (Arithmetic Progression)

In arithmetic sequences, also called arithmetic progressions, the difference between one term and the next one is constant, and you can get the next term by **adding** the constant to the previous one. If the difference is positive, it is an increasing sequence, otherwise it is a decreasing one. A simple example is 1,2,3,4... 99, 100 which is a subsequence of the integer numbers, in which the difference between one term and the next is 1. Another sequence is 1, 3, 5, 7... in which the difference is 2. The general form of such a sequence is {a, a+d, a+2d, a+3d, ... }, where **d** is the difference.

The rule for an arithmetic sequence is **x _{n} = a + d(n-1)**.

## Geometric Sequence (Geometric Progression)

In geometric sequences, also called geometric progressions, each term is calculated by **multiplying** the previous term by a constant. In a decreasing geometric sequence, the constant we multiply by is less than 1, e.g. 0.5. A general representation of a geometric progression is {a, ar, ar^{2}, ar^{3}, ...}, where **r** is the factor between the terms (common **ratio**).

The rule for a geometric sequence is simply **x _{n} = ar^{(n-1)}**.

## How to calculate n-th term of a sequence?

For an arithmetic sequence, the nth term is calculated using the formula **s + d x (n - 1)**. So the 5-th term of a sequence starting with 1 and with a difference (step) of 2, will be: 1 + 2 x (5 - 1) = 1 + 2 x 4 = 9.

For a geometric sequence, the nth term is calculated using the formula **s x s ^{(n - 1)}**. The 5-th term of a sequence starting with 1 and with a ratio of 2, will be: 1 x 2

^{4}= 16.

## Calculating the sum of an arithmetic or geometric sequence

The sum of an arithmetic progression from a given starting value to the nth term can be calculated by the formula:

**Sum _{(s,n)} = n x (s + (s + d x (n - 1))) / 2**

where **n** is the index of the n-th term, **s** is the value at the starting value, and **d** is the constant difference. For example, the sum from the 1-st to the 5-th term of a sequence starting from 1 with a step of 2 will be 5 x (1 + (1 + 2 x (5 - 1))) / 2 = 5 x 5 / 2 = 25.

The sum of a geometric progression from a given starting value to the nth term can be calculated by the formula:

**Sum _{(s,n)} = s x (1 - d^{n} / (1 - d)**

where **n** is the index of the n-th term, **s** is the value at the starting value, and **d** is the constant difference.

The above formulas are used in our sequence calculator, so they are easy to test.

## The Fibonacci Sequence

The Fibonacci sequence is a special progression with a rule of **x _{n} = x_{n-1} + x_{n-2}**. Each term depends on the previous two terms, not just the previous one. For example, having the numbers 2 and 3, the next number will be 2 + 3 = 5. The first numbers of the Fibonacci sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

This sequence is interesting as it is observed in real life natural structures, and an indefinite run of divisions of each member of the sequence by the previous (1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.66) converges to the golden ratio: 1.615... A shell's spiral follows the same form as the one drawn from a Fibonacci sequence, and it can be found in the number of petals and leaves on trees and flowers, the number of seed heads and the spiral figures they form.

#### Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation:

Georgiev G.Z., *"Sequence Calculator"*, [online] Available at: https://www.gigacalculator.com/calculators/sequence-calculator.php URL [Accessed Date: 13 Jun, 2021].

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