All the numbers can be written in a sequence defined by a certain rule where the occurrences of the numbers are defined by the rule. And there are really some special number sequences which have been a keen interest of the mathematicians. In the following article we will discuss in detail about the topic special number sequences.
More about Special Number Sequences:
There are various number sequences and few of them are quite special and some of these special number sequences are,
1. Geometric sequences.
2. Arithmetic sequences.
3. Polygonal number sequences.
4. Fibonacci number sequences.
All these sequence has a rule that defines the occurrence of the numbers in the sequence.
In the geometric sequences the consecutive numbers are formed by the multiplication of a common value to the previous numbers.
1, 4, 16, 64, 256, 1024 …
In arithmetic sequence the next number is formed by adding a value common to the whole sequence.
1, 4, 7, 10, 13, 16, 19, 22…
The Fibonacci number sequence is formed by a rule where the next number in the sequence is formed by the addition of the previous two numbers in the series,
`F_n = F_(n-1)+F_(n-2)`
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
These are the numbers which when represented as dots can be arranged to form a particular polygonal shape. So the next number in the particular polygonal shape formed by extending two consecutive arms of the polygon formed by the previous number and then filling the rest of the areas to get the perfect enlarged polygonal shape.
The numbers represented in dots can be arranged to form a triangular shape.
1, 3, 6, 10, 15…
Square number sequence:
These numbers represented in dots forms a square shape.
1, 4, 9, 16, 25, 36, 49, 64…
These are the numbers that forms the hexagon if the number is represented in dots and the next number in the sequence is obtained by increasing one dot in consecutive arms of the hexagon and then filling the rest of the hexagonal arms to get the complete hexagonal figure.
1, 6, 28, 45, 66…
These numbers are similar to the hexagonal numbers but the numbers represented as dots forms a octagonal shape.
1, 8, 21, 40, 65…