Generating Sequences By Rules

Generating sequences by rules. Another way of generating a sequence is to follow some rule relating the terms. Thus the sequence 1, 5, 9 ,13, 17... can be generated by the rules: . Start with 1. . Generate the next term by adding 4 to the term before it.

Similarly the sequence 3, 6, 12, 24, 48, .... can be generated by the rules:

. Start with 3. . Generate the next term by doubling the term before it

One disadvantage of this definition is that it is not always easy to deduce a formula for the nth term.

Example (8):

Generate the first ten terms of the sequence generated by the following rule:

. The first two terms are 1 and 1. . Generate each further term by summing the two previous ones.

(Do not attempt to find a general formula for the nth term !)

The third term is 1 + 1 or 2, the fourth one is 1 + 2 or 3 and the fifth one is 2 + 3 or 5.

The sequence thus goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ........

Aside: This is the 'Fibonacci' sequence, named after a 13th - Century mathematician, Leonardo of Pisa, 'Fibonacci - son of Bonaccio' who first investigated it.