All you need to know about Number Sequences

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Tests of the numerical sequences type are always based on a standard form. A row of numbers is presented of which you have to find the missing number. This may be at the beginning or middle of the sequence, but is usually at the end. By cleverly using mathematical actions like subtraction, addition, division and multiplication you should be able to solve the sequence and find the missing number. The numbers presented in these sequences can be either integer sequences or rational sequences.

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Integer Number Sequences

 
The first type of numbers presented in number sequences is integer number sequences, which are a form or real numbers. As the word already indicates, integer stands for incorruptible and thus series of integer numbers consist of whole numbers without fractions or decimals. When these numbers are positive integer numbers like 0, 1, 2, 3 etc they are called natural numbers, when they are negative integer numbers like -1, -2, -3 etc they are called non natural numbers. Both these types of numbers can be present in integer number sequences, resulting in a sequence like the following:

-1,     1,     3,     5,     7,     …

 
Next to the division of natural and non natural numbers, a second division can be made using the term explicit and implicit descriptions. Explicit number sequences can easily be solved by giving the sequence a formula, like the sequence shown above. The formula for this sequence is “2n−1″ for the nth term, meaning that you can chose any integer number for the letter “n” in the formula and it will generate a number in the sequence, for instance: n=3 will generate 2*3-1 = 5 as shown in the example.

An implicit number sequence is given by a relationship between its terms. For example, the Fibonacci sequence as shown below:

 
This number sequence is formed by starting with 0 and 1 and then adding any two previous terms to obtain the next one. The relationship between the numbers is called an implicit description, since you cannot define this in such an easy formula with only one variable as in an explicit definition.

Rational Number Sequences

 
Unlike integers, rational numbers are numbers which can be written as a fraction or quotient where numerator and denominator both consist of integers, meaning that top and bottom of the fraction are whole numbers. Rational numbers can also be written by decimal expansion which either terminates after a finitely amount of numbers or repeats the same sequence over and over. Examples of rational numbers are ½, ¾, 1.75 and 3.25.

Next to rational numbers, also irrational numbers exists. These sequences consist of real numbers which cannot be expressed as a fraction, but only via expansion in decimals. Even then the decimals are not terminated after a finite amount of numbers but continue without repetition of the sequence. Examples of irrational numbers are the square root of 2, pi and e.

As explained above number sequences exist in many forms and types. In order to improve your numerical reasoning skills it is best to practice all these different types and forms in order to master them.

Number Sequences Examples & 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. On his page the most common number sequences examples are presented.

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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:

X_n = x₁ + (n – 1)d

An example of this type of number sequence could be the following:

3,  8,  13,  18,  23,  28,  33,  38,  …

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:

25,  23,  21,  19,  17,  15,  …

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:

X_n= x₁ r ^n-1

An example of this type of number sequence could be the following:

2,  4,  8,  16,  32,  64,  128,  256,  …

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:

2187,   729,   243,   81,   27,   9,   3,   …

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:

X_n = (n² + n) / 2

An example of this type of number sequence could be the following:

1,   3,   6,   10,   15,   21,   28,   36,   45,   …

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:

X_n= n²

An example of this type of number sequence could be the following:

1,   4,   9,   16,   25,   36,   49,   64,   81,   …

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:

X_n= n³

An example of this type of number sequence could be the following:

1,   8,   27,   64,   125,   216,   343,   512,   729,   …

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
F_n = F_n-1 + F_n-2

An example of this type of number sequence could be the following:

0,   1,   1,   2,   3,   5,   8,   13,   21,   34,   …

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).