Lesson 2:
Recursive Sequences
Student Outcome:
Students write recursive
and explicit formulas for
sequences.
Recursive Sequence:
-a sequence where a previous
term or terms in the sequence is
used to find the next one
Recursive Sequences
An example of a recursive
sequence would look like this:
an =an-1+1 where a1=1
Recursive Sequences
Now this notation may look
confusing but simply plug values
in for the n to find the next terms
in the sequence.
an =an-1+1 where a1=1
Recursive Sequences
an =an-1+1 where a1=1
Since we know a1, we need to
now find the next terms a2 by
plugging 2 in for n.
a2=a2-1+1
Recursive Sequences
Now simplify...
a2=a1+1
We know what a1=1, so we
substitute 1 in for a1.
a2=1+1
a2=2
Recursive Sequences
Now find the terms a3 and a4
given that you now know a1=1
and a2=2.
Recursive Sequences
an =an-1+1 where a1=1 and a2=2.
a3=a3-1+1
a3=a2+1
a3=2+1
a3=3
Recursive Sequences
an =an-1+1 where a1=1, a2=2, and
a3=3.
a4=a4-1+1
a4=a3+1
a4=3+1
a4=4
Recursive Sequences
On your own find the first 5
terms of the following recursive
sequence.
an=2an-1+3 where a1=2
Recursive Sequences
an=2an-1+3 where a1=2
a1=
a2=
a3=
a4=
a5=
Recursive Sequences
Some recursive sequences can
be expressed in different ways
and sometime involve multiple
terms but the process is exactly
the same.
Recursive Sequences
For example find the first four
terms of this sequence…
an+1=an+2 given a1=3
Recursive Sequences
an+1=an+2 given a1=3
a2=a1+1= a1+3
now plug 3 in for a1…
a2=3+3=6
Recursive Sequences
an+1=an+2 given a1=3 and a2=6
so...
a3=a2+1=a2+2
now plug in 6 for a2…
a3=6+2=8
Recursive Sequences
an+1=an+2 given a1=3, a2=6, and a3=6
On your own find a4 and a5…
a4=
a5=
Recursive Sequences
Ex) Given the recursive sequence below find the
next 4 terms.
an+1=-2an-3 given a1=-3
a1=-3
a2=
a3=
a4=
a5=
Recursive Sequences
Ex) Given the recursive sequence below find the
next 4 terms.
an+1=-2an-3 given a1=-3
a1= -3
a2= 3
a3= -9
a4= 15
a5= -33
Recursive Sequences
Other recursive sequences will sometimes
involve using multiple terms like the famous
Fibonacci sequence…
an=an-1+an-2 given a1=1 and a2=1
So the next few terms would be as follows…
a3= a3-1+a3-2= a2+a1= 1+1= 2 so a3= 2
a4= a4-1+a4-2= a3+a2= 2+1=3 so a4= 3
Recursive Sequences
an=an-1+an-2 given a1=1, a2=1, a3=2, and a4=3
Now on your own find the next two terms of the
Fibonacci Sequence…
a5=
a6=
Recursive Sequences
Ex) Given the first two terms of the recursive
sequence below find the next three terms.
an= 2an-1- 3an-2 given a1=2 and a2=-2
a3=
a4=
a5=
Recursive Sequences
Ex) Given the first two terms of the recursive
sequence below find the next three terms.
an= 2an-1- 3an-2 given a1=2 and a2=-2
a3= -10
a4= -14
a5= 2