8.1 Sequences
Here are two examples of
.
Sequence 1:
3, 6, 9, 12, 15
Sequence 2:
3, 6, 9, 12, 15,…
The numbers in the sequences are called
Sequence 1 is called a
.
sequence because it has a last term.
Sequence 2 is called an
An
fixed number, called the
term.
sequence because it continues without stopping.
sequence is any sequence in which the first is obtained by adding a
, to the preceding
Questions 1-3: State whether each of the following is an arithmetic sequence. If it is, give the next term.
1. 4, 1, -2, -5,…
2. 1, 4, 9, 16,…
3.
1
7
, 2, , 5,...
2
2
There are two different ways to write the equation for a sequence. In either formula, the n represents the
term number. We use subscript to denote a specific term. For example, a1 is the first term of the
arithmetic sequence and a n is the nth term of the arithmetic sequence.
Recursive Formula: Recursive formulas give
, which is the
sequence and then give a rule or formula for subsequent terms using the
to find the next.
Example:
a1  7 ,
term of the
term
an 1  an  8
For arithmetic sequences, we can write the recursive rule as
difference.
an1  an  d where d is the common
Questions 4-6: State the second and third term of the arithmetic sequence whose first term is given.
4.
a1  8, d  6
5.
1
a1   , d  1
2
6.
a1  4, d  
1
2
Questions 7-8: State the first four terms of the sequence. If it is arithmetic, give the common difference.
7.
a1  6, an1  3an  4
8.
a1  12, an 1  4  an
10.
6, 12, 24, 48,…
Questions 9-10: Specify each sequence recursively.
9.
13, 7, 1, -5,…
Explicit Formula: A sequence can be specified by giving the value of any term as a function of its
in the sequence, n.
Example:
an  3 2 n
We will learn rules for explicit formulas for specific types of sequences in future lessons.
Questions 11-12: State the first four terms of the sequence. If the sequence is arithmetic, give the
common difference.
11.
an  3n  1
12.
an  3 2 n1
14.
1, 4, 9, 16,…
Questions 13-14: Specify each rule explicitly.
13.
2, 6, 10, 14,…