# Number Sequence Calculator

## Solve arithmetic, geometric, and Fibonacci sequences

The sequence solver calculator works with three common sequences: arithmetic progression, geometric progression, and Fibonacci sequences. Using this calculator you will be able to find any number in a sequence of the given methods – this can be helpful for students checking their math work, and anyone looking to improve their numerical reasoning. Mastering this skill can help you make accurate loan estimations, keep track of business costs, and may be helpful in job aptitude tests.

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What is a number sequence?
A number sequence, as the name suggests, is a row of numbers that follow a pattern and are often presented with a number missing, or a number required that is not shown in the progression. Number sequences are typically used as a measure of intelligence on IQ and aptitude tests, when combined with other test types and formats as well. Working out the missing number requires use of basic math skills such as addition, subtraction, multiplication, or division to find out the pattern of the sequence.
Arithmetic progression
In mathematics, an arithmetic progression is when the numbers in a sequence have a difference that is constant throughout the terms.
For example, the sequence 5, 10, 15, 20, 25 has a common difference of 5.
While, the sequence 20, 22, 24, 26, 28… has a common difference of 2.
And 9, 6, 3, 0, -3, has a common difference of -3.
When the progression has a clear beginning and end, like example one and three above, it is called a finite arithmetic progression and the sum of that progression is called an arithmetic series. The progressions may otherwise grow positive or negative to infinity as denoted by any dots at the end, depending on the common difference- this is shown in example two.
AP properties and formulas
When it comes to AP the properties can be defined as,

### a, a+d, a+2d, a+3d, a+4d, …

Where:
A – refers to the first term in the sequence
D – refers to the common difference
Example :
 1, 3, 5, 7, 9, (d= 2) 1( a ), 1+2( d ), 1+2×2( d ), 1+3×2, 1+4×2,
By manipulating the properties of an AP you can easily find any number in the sequence, without having to write out the whole sequence. The formula the find an nth value, an, is:

### an = a + (n-1)d

The values in this formula are:
a = the first term of the arithmetic progression, the starting point.
d = the common difference between the terms.
n = the number of terms in the arithmetic progression.
an = the nth value.
Why? Well, to find the value of the nth term, an, you must multiply the previous number in the sequence, (n-1), by the difference, d, and add it on to the starting point, a.
Example: Given an arithmetic progression with the initial term 5 and the common difference 7, what is the 10th term?
an = 5 + (10-1)7
an = 5 + (9)7
an = 5 + 63
an = 68
Therefore, the 10th term has a value of 68.
Try out these practice questions and check your work using the sequence solver:
1. Find the 15th term of the AP with the first term 8 and the common difference 2
2. Write down the 10th and 19th terms of the AP 8, 5, 2…
Geometric progression
In mathematics a geometric progression is when the numbers in a sequence are multiplied by a constant number known as the common ratio. Each new term after the first is found by multiplying the previous one by the constant.
As an example 2, 6, 18, 54… has a common ratio of 3
While, 1, -2, 4, -8… has a common ratio of -2
And 1, 0.5, 0.25, 0.125 has a common ratio of ½
Like an arithmetic progression a geometric progression can be finite as in example three, or infinite as in example one and two. Therefore, the sum of a finite geometric progression is called a geometric series.
GP properties and formulas
When it comes to a GP the properties can be defined as:

### * signifies exponent

Where,
a is the first term in the progression, and
r is the common ratio
Example:
 2 6 18 54 2( a ) 2×3( r ) 2×3*2 2×3*3
So then, manipulating the properties of a GP makes it very easy to find any number in the sequence. The formula the find an nth value, an, is:

### * denotes exponent

a = the first term of the geometric progression, the starting point.
r = the common ratio between the terms.
n = the number of terms in the geometric progression.
an = the nth value.
Why? Well, to find the value of the nth term, an, you must exponentially multiply the previous number of the sequence, (n-1), by the ratio, r, and multiply that into to the starting point, a.
Fibonacci Sequence, spirals and the golden ratio
The Fibonacci sequence is another mathematical pattern that is characterized by every number being the sum of the two preceding ones. This sequence is also known as the golden ratio and has a much deeper universal meaning then the other progressions reviewed here as it is seen extensively in mathematics and in nature and biology.
Fibonacci numbers first appeared first in old Indian math Sanskrit but gained popularity when appearing in a book called Liber Abaci by Leonardo Pisano, also known as Fibonacci. The author developed the sequence by considering the growth of a theoretic rabbit population in a field, which, followed certain circumstances and rules:
1. The story must begin with one pair of rabbits in a field.
2. Each pair of rabbits is considered a count of one.
3. All rabbits are able to mate at the age of one month.
4. So by the second month the female can produce a new pair of male and female rabbits.
5. The rabbit pairs would always mate at their first month of age and always produce a new pair of males and females from the second month onwards.
6. The rabbits can never die.

### Following these rules it stands to reason that:

• At the end of the first month, there would still only be one pair of rabbits.
• Then during the second month the female would produce her first litter making a second pair of rabbits.
• During the third month she would produce yet another litter making three pairs of rabbits while the previous generation mates for the first time at the age of 1 month.
• Once the fourth month comes along the original female would produce another pair, while the female born two generations ago would produce her first pair, effectively making five pairs of rabbits.
• And so the story continues…
Fibonacci sequence properties and formulas
Taking the rabbits our of the equation and looking at just the numbers, the sequence becomes: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …
Because each term is then sum of the two terms berfore, the properties can be derived as follows,

### Fn= Fn-1 + Fn-2

Yet, in order to find an nth number in the series it gets slightly more difficult and requires more knowledge on the Fibonacci sequence. To understand more lets start by taking the ratio of two successive numbers in the Fibonacci sequence and dividing each number from the third onwards by the number before.
For example:
144/89= 1.61797…
233/144= 1.61805
The number that appears through the division of the successive terms eventually settles down to a particular one: 1.618034. This number is known as the golden ratio, golden mean, or simply the golden number, and is signified by the Greek letter Phi (ϕ). With knowledge of the common ration between terms, phi, and using it’s reciprocal phi, it is possible to find the nth number in the series. The formula, known as Binets formula, is as follows:
Fn= Phin – (-Phi)n
√5
It is a surprising formula because it involves square roots and powers of Phi, but the formula always produces an integer. However, this formula may be limiting based on the amount of decimal places that are being computed.
Try some practice questions using the formula and check your work with the number sequence calculator.