Sequences

A sequence is an ordered list of numbers or objects arranged according to a rule.

A sequence of numbers is a function s: NR

Each number of the sequence is called a term, and we represent it as ai. We call general term of a sequence, an, the expression which represents any term of the sequence.

For example:

· 1, 2, 3, 4, 5, … an = n, then a100 = 100

· 2, 4, 6, 8, 10, … an = 2n, then a25 = 50

· 1, 4, 9, 16, 25, … an = n2, then a12 = 144

· 32, 16, 8, 4, 2, … an = 26-n, then a10 = 2-4 = 1/16

Some sequences are recurring, because we obtain each term from the previous one. For example:
· 1, 1, 2, 3, 5, 8, 13, … an = an-1 + an-2     a1 = a2 = 1

This is the Fibonacci sequence.

An arithmetic progression is such a sequence of numbers that the difference of any two successive members of the sequence is a constant. The constant is called difference of the progression.
Example: 3, 5, 7, 9, …. d = 2
General term: an = a1 + (n-1)· d
In the example: an = 3 + (n-1)· 2 = 2n + 1

A geometric progression is such a sequence of numbers that the quotient of any two successive members of the sequence is a constant, which is called ratio of the progression.
Example: 3, 6, 12, 24 …. r = 2
General term: an = a1· rn-1
In the example: an = 3 · 2n-1

Exercise: find the general term and a10 of these sequences:

a) 1, 3, 9, 27, 81, ...

b) 15, 12, 9, 6, 3, 0, ...

c) 1, -2, 3, -4, 5, ... Solutions: a) an=3n-1, a10= 39= 19683; b) an= 18- 3n, a10=-12; c) (-1)n·n; a10= 10; 