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Special groups of numbers

By arranging numbers in certain ways, various patterns can be formed. This chapter looks at the following patterns:

  • figurate numbers
  • palindromic numbers
  • Fibonacci numbers
  • Pascal's triangle.

Figurate numbers

Figurate numbers are numbers that can be organised into a regular geometrical arrangement.

Figurate numbers form a certain pattern.

For example:

1
1 + 2
1 + 2 + 3
1 + 2 + 3 + 4
1 + 2 + 3 + 4 + 5

When this pattern is represented geometrically, in the form of dots, it creates a triangular formation:

= 1

= 1+2

= 1+2+3

=1+2+3+4

=1+2+3+4+5

1

3

6

10

15

A similar pattern can be formed using square numbers:

1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1 + 3 + 5 + 7 + 9


=1

=1+3

=1+3+5

=1+3+5+7

=1+3+5+7+9

1

4

9

16

25

Palindromic numbers

Palindromic numbers are numbers that read the same forwards and backwards.

There are an infinite number of palindromic numbers. All one-digit numbers are palindromic. Many (but not all) factors of 11 are palindromic.

Here is a list of some palindromic numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33 ... 99, 101, 111, 121, 131..... 202, 212, 222, 232, 242 .....

It's possible to create a palindromic number by taking any multi-digit number and adding itself to itself in reverse.

For example:

If the first answer is not a palindrome, simply repeat the process until a palindromic number is the result.

For example:

Fibonacci numbers

Fibonacci numbers are sequences of numbers developed by adding the previous two numbers in the series. The first two numbers of the series are both 1.

The set of Fibonacci numbers includes:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377

Where,

1 = 1
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89
55 + 89 = 144
89 + 144 = 233
144 + 233 = 377

Pascal's Triangle

Pascal's Triangle is a series of continually lengthening number sequences. Each number within these rows is the sum of the two numbers immediately above it.


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Chapters: Special groups of numbers

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Question 1/5

1. Blaise Pascal worked with patterns of numbers contained in the shape of:-

a pentagon

a triangle

a square

a circle

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