9.1 Sequences and Series

=An infinite sequence is a function whose domain is the set of positive integers. The function values

a1, a2, a3, a4, … , an

are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

There are two types of formulas to display these sequences: Recursive and Explicit.

For this sequence: -5, -2, 1, 4, 7, 10 here are the recursive and explicit formulas.

Recursive:
a1 = -5
ak+1 = ak+3

k represents the next term for the sequence. It shows how to find out the next term.

Explicit:

an = 3n-8

n represents the current term for the sequence and how to find it. For example, the first term (1) is found by multiplying 1 by 3 and then subtracting by 8. You can do this for each term following the first term with 2, 3 ,4 and so on.

Here are some other examples of some basic sequences:

n² = 1, 4, 9, 16, 25, 36, …
n³ = 1, 8, 27, 64, 125, …
2^n = 2, 4, 8, 16, 32, …
3^n = 3, 9, 27, 81, …
n! = 1, 2, 6, 24, 120, ...
! represents factorial. If n is a positive integer, n! = 1 · 2 · 3 · 4 · ... · (n-1) · n.

0!=1
1!=1
2! = 1 · 2 = 2
3! = 1 · 2 · 3 = 6
4! = 1 · 2 · 3 · 4 = 24
5! = 1 · 2 · 3 · 4 · 5 = 120