standard_description: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1.
problem_activity: Problem 1
Consider the sequence below.
$${1, \space1,\space 2,\space 3,\space 5,\space 8,\space 13,\space 21, \space34, …}$$
a.   Describe the pattern that you notice. How is each next term determined?
b.   This pattern is famously called the Fibonacci sequence. Write a recursive formula to represent the Fibonacci sequence. Write a formula in sequence notation using
$${a_n}$$
and in formula notation using
$${f(n)}$$
.
c.   If
$${a_{15}=610}$$
and
$${a_{16}=987}$$
, what are the values of
$${a_{17}}$$
and
$${a_{14}}$$
?
Problem 2
For each sequence below, write a recursive formula, in both sequence notation and function notation, to represent the sequence.
a.
$${18,\space 13,\space 8,\space 3,\space -2,\space …}$$
b.
$${4,\space 12,\space 36,\space 108,\space 324,\space …}$$
Problem 3
A sequence is given by an explicit formula, as shown below, where
$$n$$
represents the term number.
$$a_n=4n-1$$
for
$$n\geq1$$
a.   Write the first five terms in the sequence.
b.   Write a recursive formula to represent the sequence.


aligns: yes