Download 1.1 Recursively defined sequences PP

Ch. 1 Patterns and Recursions
Recursively Defined Sequences.
Note: This is similar to recursions in Algebra I, where
we used {0,start} {ans(1)+1, ans(2)+rule}
We will use different notation.
Learning Targets:
 I can write recursive formulas for generating
sequences using proper notation given a list of
numbers or a real-world scenario.
 I can use a recursive formula to generate a
sequence.
 I can distinguish between arithmetic, geometric and
shifted geometric sequences given a formula, list of
numbers, or graph.
Recursion- a process in which each
step of a pattern is dependent on the
step or steps that come before it.
Note: Arches in a cathedral show artistic use of
repeated patterns, scientists use patterns and
repetition to conduct experiments, gather data, and
analyze results.
Example A (page 28)
Discuss with a partner…
1) How many people can sit at 10 tables
arranged in a straight line?
2) How many tables are needed to seat 32
people?
You have 2 minutes…
Sequence- ordered list of numbers.
Term- each number in the sequence.
 The 1st term is defined as u1 (read as: “u sub 1”)
General term- the nth term in a sequence.
 Written as un (read as: “u sub n”)
Recursive formula- the formula that defines a
sequence, must specify one (or more) starting values
and a recursive rule.
Recursive rule- defines the nth term in relation to
the previous term.
Usually we start with the 0th term or the 1st term,
but you can really start at any term you like. Then
just say how to get from one term to the next after
that. This is what is meant by the rule.
Back to Example A…
Sequence: 4, 6, 8, 10…
Recursive Formula:
u1 = 4
un = un-1 + 2
where n ≥2
This means the 1st term is 4.
This is saying to get to the next
term (the nth term), you take
the previous term (the n-1 term)
and add 3 to it!
Because we already know n=1
(the 1st term)
In Algeba I, we would have written: {1,4} {ans(1)+1, ans(2)+2}
Example:
10, 6, 2, -2, -6, -10…
What is the 1st term?
What is the rule?
Then the recursive formula would be:
u1 = 10
un = un-1 – 4 Where n ≥ 2
An arithmetic sequence is a sequence in which
each term is equal to the previous term plus a
constant.
Note: The constant can be positive or negative.
The constant is called the common difference.
un= un-1+ d
An geometric sequence is a sequence in which
each term is equal to the previous term multiplied by
a constant.
This constant is called the common ratio.
Common ratio:
un
r
un 1
The constant is called the common difference.
un= r * un-1
Example:
3, 6, 12, 24…
What is the 1st term?
What is the rule?
Then the recursive formula would be:
u1 = 3
un = 2∙un-1
where n ≥ 2
What is the 10th term?
Example C (page 32)
Discuss with partner…
1) Use the picture to write the first 4 terms.
2) Write a recursive formula. (Careful with your 1st term)
3) How many triangles will be shaded in the 20th stage?
You have 3 minutes…
Homework:
Page 34 1-5
Ch. 1 Patterns and Recursions
Recursively Defined Sequences.
More practice of what we learned yesterday…
today we learn how to use the notation in our
calculators!!
Learning Targets:
 I can write recursive formulas for generating
sequences using proper notation given a list of
numbers or a real-world scenario.
 I can use a recursive formula to generate a
sequence.
 I can distinguish between arithmetic, geometric and
shifted geometric sequences given a formula, list of
numbers, or graph.
 I can use technology to simulate arithmetic and
geometric sequences.
Calculator Notes:
Mode: Sequence
y=
Line 1: nMin = 1
Line 2: u(n) =
Line 3: u(nMin) =
Don’t change.
Rule
Starting number
Home Screen: Enter u(__) to find specific value
Table: Move through list to find specific value
Example:
5, 6.5, 8, 9.5, …
Then the recursive formula would be:
u1 = 5
un = un-1 + 1.5
where n ≥ 2
Use your calculator to find the 25th term.
Example:
2, 6, 18, 54, …
Then the recursive formula would be:
u1 = 2
un = 3∙un-1
where n ≥ 2
Use your calculator to find the 12th term.
Example: A swimming pool has a small hole and is
draining water at a rate of 0.13 gallons per hour.
When your little brother gets done swimming, the pool
has 8 gallons of water in it.
a) How much water will be in the pool if your brother
wants to swim again the following afternoon (23
hours later).
b) If no one uses the pool, how long will it take for all
of the water to drain?
Example: I deposited $1,000 into a saving account
that earns 2.3% interest each month. I am not
making any more deposits or withdraws from the
account.
a) How much money will be in my account in 1
year (12 months)?
b) How long will it take me to double my money?
Homework:
Page 34 7 – 9, 11, 12