PRECALCULUS CHAPTER NINE
Sequences
Vocabulary & Notation:
Sequence: an ordered progression of numbers.
Term: each number in a sequence is a term
First term is a1
Second term is a2
nth term is an
The subscripts only denote the position of the term in the sequence.
Types of Sequences:
Arithmetic: a sequence in which there is a common difference between every pair of
successive terms. Example: 5,8,11,14
Geometric: a sequence in which there is a common ratio (or quotient) between every pair
1 1 1 1
of successive terms. Example: , , , ,...
2 4 8 16
1 1 1 1
Infinite: there is an infinite number of terms in the sequence. Example: , , , ,...
2 4 8 16
Finite: there is a finite number of terms in the sequence. Example: 5,8,11,14
Sequences are infinite unless otherwise specified.
Explicitly Defined Sequence:
A formula is given for any term in the sequence.
Example: ak = 2k -5 . find the first 5 terms and the 20th term in the sequence.
Recursively Defined Sequence:
The first term is given and a rule to obtain each succeeding term from the one preceding it.
Example: b1 = 8 and bn = bn-1 - 2 for all n >1 . Find the next 4 terms in the sequence.
General formulas for finding terms in a sequence:
Arithmetic: an = a1 + (n -1)d
( )
Geometric: an = a1 r n-1
To use these:
1) Determine if the sequence is arithmetic or geometric
2) Find the common difference or common ratio.
3) Substitute into the general formula (simplify the arithmetic formula)
Example 1: Find the 20th term of the sequence 55, 49, 43…. Then write a recursive and
explicit rule for the sequence.
Example 2: Find the 8th term of the sequence
explicit rule for the sequence.
1 4 16
, , ,... Then write a recursive and
5 5 5