Number Sequences – Examples and Types
Number sequences consist of a finite row of numbers of which one of the numbers is missing in the sequence. As the term sequence already indicates, it is an ordered row of numbers in which the same number can appear multiple times. Within these number sequences a lot of variety exists of which the most commonly known will be presented here.
Arithmetic Sequences
An arithmetic sequence is a mathematical sequence consisting of a sequence in which the next term originates by adding a constant to its predecessor. When the first term x1 and the difference of the sequence d is known, the whole sequence is fixed, or in formula:
An example of this type of number sequence could be the following:
This sequence has a difference of 5 between each number. The pattern is continued by adding the constant number 5 to the last number each time. The value added each time is called the “common difference”. The common difference could also be negative, like this:
This common difference is -2. The pattern is continued by subtracting 2 each time.
Geometric Sequences
A Geometric sequence is a mathematical sequence consisting of a sequence in which the next term originates by multiplying the predecessor with a constant, better known as the common ratio. When the first term x1 and the common ratio r are known, the whole sequence is fixed, or in formula:
This sequence has a factor of 2 between each number, meaning the common ratio is 2. The pattern is continued by multiplying the last number by 2 each time. Another example:
This sequence has a factor of 3 between each number, however as can be seen the sequence can work both by increasing as well as decreasing the value of numbers. The pattern is continued by dividing the last number by 3 each time.
Special Number Sequences
Triangular Numbers
Triangular numbers fall into the category of polygonal numbers of which the last represents the number connected to the amount of dots presented in the figure. In the case of triangular numbers these dots represent the amount of dots needed to fill a triangle, starting with the smallest number possible, or in formula:
An example of this type of number sequence could be the following:
This sequence is generated from a pattern of dots which form a triangle. By adding another row of dots and counting all the dots we can find the next number of the sequence.
Square Numbers
Square numbers, better known as perfect squares, are an integer which is the product of that integer with itself. Square numbers are never negative and thus the square root of a square number is always an integer, or in formula:
An example of this type of number sequence could be the following:
The sequence consists of repeatedly squaring of the following numbers: 1, 2, 3, 4 etc. since the 10th number of the sequence is missing, the answer will be 102 = 100.
Cube Numbers
A cube number sequence is a mathematical sequence consisting of a sequence in which the next term originates by multiplying the number 3 times with itself, or in other words, raising it to the power of three, in formula:
An example of this type of number sequence could be the following:
The next number is made by cubing in this case the 10th number and thus 103 = 10*10*10= 1000.
Fibonacci Numbers
A Fibonacci number sequence is a mathematical sequence consisting of a sequence in which the next term originates by addition of the previous two. By definition the first two numbers of this sequence are 0 and 1, after which the subsequent numbers can be calculated, of in formula for n>1:
F0 = 0
F1 = 1
Fn = Fn-1 + Fn-2
An example of this type of number sequence could be the following:
The next number is found by adding the two previous numbers. The 2 is found by adding the two numbers in front of it (1+1). The 21 is found by adding the two numbers in front of it (8+13). The next number in the sequence above would be 55 (21+34).
Related Pages
