Download Note Outlines (1.2: Number Patterns)

Section 1.2: Number Patterns
Definition: A sequence is an ordered list of numbers.
The numbers in this list are called terms.
Our goals will be to
 find a pattern in a sequence
 use that pattern to determine the type of the sequence
 predict the next term.
Arithmetic Sequences
Example. Determine the next term of the sequence
3, 11, 19, 27, ...
The sequence 3, 11, 19, 27, ... is an arithmetic sequence
because we can get from one term to the next by adding
(or subtracting) the same value.
This value is called the common difference, d.
Geometric Sequences
Example. Determine the next term of the sequence
2/9, 2/3, 2, 6, ...
The sequence 2/9, 2/3, 2, 6, ... is a geometric sequence
because we can get from one term to the next by
multiplying (or dividing) by the same value.
This value is called the common ratio, r.
Other Sequences
Not every sequence is arithmetic or geometric.
Example. Determine the next term of the sequence
8, 14, 22, 32, ...
Example. Determine the next term of the sequence
1, 1, 2, 3, 5, ...
Finding the nth Term (General Term) of a Sequence
The nth term of an arithmetic sequence:
an  a1  n  1d
Example. Recall the sequence 3, 11, 19, 27, ...
(a) Find the formula for the general term (the nth term).
Example. Recall the sequence 3, 11, 19, 27, ...
We’ve seen that the general term is 𝑎𝑛 = 8𝑛 − 5.
(b) Find the value of the sixth term.
The nth term of a geometric sequence:
a n  a1  r n 1
Example. Recall the sequence 2/9, 2/3, 2, 6, ...
(a) Find the formula for the general term (the nth term).
Example. Recall the sequence 2/9, 2/3, 2, 6, ...
2
We’ve seen that the general term is 𝑎𝑛 = × 3𝑛−1 .
9
(b) Find the value of the sixth term.
Example: For each sequence,
(i) Determine if it is arithmetic, geometric, or neither.
(ii) Give the next two terms in the sequence.
(iii) If the sequence is arithmetic or geometric, write a
formula for the general term (the nth term).
(a) 10, 13, 16, 19, ...
(b) 2, 5, 7, 12, 19, ...
(c) 1.5, 4.5, 13.5, 40.5, ...
(d) 98, 97, 95, 92, ...