RECURSIVE FORMULAS
Focus 7 Learning Goal –
(HS.F-BF.A.1, HS.F-BF.A.2, HS.F-LE.A.2, HS.F-IF.A.3)
=
Students will build a function (linear and exponential) that models a relationship
between two quantities. The primary focus will be on arithmetic and geometric
sequences.
4
3
2
1
0
In addition to level
3.0 and above and
beyond what was
taught in class, the
student may:
· Make connection
with other concepts in
math
· Make connection
with other content
areas.
The student will build a function
(linear and exponential) that
models a relationship between
two quantities. The primary focus
will be on arithmetic and
geometric sequences.
- Linear and exponential functions
can be constructed based off a
graph, a description of a
relationship and an input/output
table.
- Write explicit rule for a
sequence.
- Write recursive rule for a
sequence.
The student will be
able to:
- Determine if a
sequence is
arithmetic or
geometric.
- Use explicit rules
to find a specified
term (nth) in the
sequence.
With help from the
teacher, the student
has
partial success with
building a function
that models a
relationship
between two
quantities.
Even with help, the
student has no
success
understanding
building functions
to model
relationship
between two
quantities.
EXPLICIT FORMULA (REVIEW)
Sequence Term
Term
a1
2
Write the explicit formula for the sequence:
2, 4, 6, 8…
a2
4
What is the pattern? How is each term related to the
a3
6
term number?
a4
8
The explicit formula is: an = 2n
An explicit formula allows you to determine any term
in a set sequence.
RECURSIVE FORMULA
A recursive formula always uses the preceding term to define the
next term of the sequence.
Write the recursive formula for the sequence:
2, 4, 6, 8…
A recursive formula tells us how each term is connected to the next
term.
The difference between each term is 2 (a1 = 2) we can display this
in a recursive formula using the following:
an = an-1 + 2
an = term number
and
an-1 = the term before the n term
HOW DOES A RECURSIVE FORMULA WORK?
an = an-1 + 2
2, 4, 6, 8…
The 4th term in this
sequence 8. (a4 = 8)
Find the 5th term.
a5 = a(5-1) + 2
a5 = a4 + 2
a5 = 8 + 2
a5 = 10
an = an-1 + 2
The 5th term in this
sequence 10. (a5 = 10)
Find the 6th term.
a6 = a(6-1) + 2
a6 = a 5 + 2
a6 = 10 + 2
a6 = 12
USE THE RECURSIVE FORMULA TO WRITE THE
ST
1 FIVE TERMS OF THE SEQUENCE.
an = an-1 – 2,
a1 = 27
We are provided the 1st term of
the sequence, 27. We need to
find the next four terms.
a2 = 27 – 2
a2 = 25
a3 = 25 – 2
a3 = 23
a4 = 23 – 2
a4 = 21
a5 = 21 – 2
a5 = 19
The first five terms of the
sequence are 27, 25, 23, 21,
and 19.
1, 1, ,2, 3, 5, 8, 13, 21, 34, …
One of the most famous sequences is
the Fibonacci sequence.
How is each term generated?
What would be the next term?
an = a(n-1) + a(n-2)
a10 = a9 + a8
a10 = 34 + 21
a10 = 55
Sequence Term
Term
a1
a2
a3
a4
a5
a6
a7
a8
a9
1
1
2
3
5
8
13
21
34
Write the first 5 terms of the sequence
using the explicit formula given.
Then, write the recursive formula for the sequence.
an = 2n + 10
Substitute the term numbers 1 through 5 for “n” to write the first 5
terms of the sequence.
12, 14, 16, 18, 20
How would you write the recursive formula?
Each term is increased by 2. Just add two to the previous term.
an = a(n-1) + 2, where a1 = 12
Why do we have to say what a1 is?
9, 1, -7, -15…
Write an explicit formula for the
sequence.
Since the sequence is subtract 8, you need
to multiply the term number by -8.
What do you need to do next in order to
get to the first number?
an = -8n + 17
Sequence Term
Term
a1
9
a2
1
a3
-7
a4
-15
9, 1, -7, -15…
Write a recursive formula for the
sequence.
an = -8n + 17
You subtract 8 to get to the next term.
a38 = -8(38) + 17
an = a(n-1) - 8
a38 = -304 + 17
Which formula would you use to find the 38th
term?
a38 = - 287
The explicit formula is best for finding specific
terms in a sequence. an = -8n + 17