5.6 Special sequences:

5.6.1

For example: 1, 3, 6, 10, 15, 21, 28, 36, 45…..

Formula used;

x_n = n(n+1)/2

5.6.2 Square numbers:

For example: 1, 4, 9, 16, 25, 36……….

The next number is made by squaring where it is in the pattern.

Formula used;

x_n = n²

5.6.3 Cube numbers:

For example: 1, 8, 27, 64, 125…………

The next number is made by squaring where it is in the pattern.

Formula used;

x_n = n³

5.6.4 Fibonacci numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21……

The next number is obtained by adding the two numbers before it together

1. i) The number 2 is obtaining by adding the two numbers before it, i.e. (1+1)
2. ii) Similarly, 3 is obtained by (1+2)
3. iii) And the 5 by (2+3)

Formula used:

x_n = x_n-1 + x_n-2

• x_n is term number "n"
• x_n-1 is the previous term (n-1)
• x_n-2 is the term before that (n-2)

In the Fibonacci Numbers the terms are numbered from 0 onwards.

5.6.4.1 Golden ratio: The golden ratio is a special number approximately equal to 1.618

If you divide a line into two parts, such that the longer part is divided by the shorter part then it is equal to the whole length divided by the longer part. This gives you the golden ratio.

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