Transcript lesson 2.2

SECTION 2-2
Finding the nth Term
• You have been looking at different sequences
• Each time you were asked to describe the next
one or two terms.
3, 5, 7, 9, 11, _?_, _?_
• Here is a geometric sequence.
• What is the height on the next two rectangles?
• What is the width of the next two rectangles?
• What is the area of the next two rectangles?
Sometimes we don’t want to know just the next
two terms, so we have to look at the sequence
differently.
We very often number each term in a
sequence.
1 2 3 4 5
3, 5, 7, 9, 11, _?_, _?_
What will the 10th term be?
1
2
3
4
5
6
What will the 12th term be?
7
8
What would you do to get the next term in the sequence
20, 27, 34, 41, 48, 55, . . .?
Add 7 would be a good approach so
the 7th term would be 55 + 7 or 62
Write down what you believe the 10th term will be?
If you got 83, try to explain how you
found the 10th term.
If you knew a rule for calculating any term in a
sequence, without having to know the previous
term, you could apply it to directly to calculate the
any term. The rule that gives the nth term is call the
function rule.
Function
Rule
Suppose the rule for a function was to double the
number. Describe what would happen to each
number as it is put in the function machine.
5
8n
1/2
3
-3
Rule is to double the input number
Function
Rule
16
-6
1
2n
10
6
Finding the Rule
• Copy and complete each table. Describe how the second
row of numbers is changing. (increasing or decreasing
and by how much)
Watch for patterns within each problem.
Did you spot the pattern?
• If a sequence has a constant difference of 4, then
the number in front of the n (the coefficient of n) is
4
______.
• If a sequence had a difference -2, then the
number in from of the n (the coefficient of n) is
-2
______.
• If a sequence had a difference 3, then the number
3
in from of the n (the coefficient of n) is ______.
If a sequence increased by a certain
amount such as 1, what was the coefficient
on the n in the rule?
If a sequence decreased by a certain amount
such as 2, what was the coefficient on the n
in the rule?
• Let’s look at one more thing about the rule.
Notice the terms numbers begin with 1. If we
could backward to term 0 what would the value
be in each table?
0
-5
0
-3
0
5
0
-2
0
7
0
-5
0
-3
0
5
0
-2
0
13
The constant difference is 7, so you know part of the
rule is 7n. How do you find the rest of the rule?
We notice that the first term is 20, but what value
matches up with 0 ?
What expression would represent the nth term?
7n + 13
Will this expression generate the other terms?
Example
• Find the rule for the sequence 7, 2, -3, -8, -13, -18
0
12
The rule is decrease by 5, so the rule must be
-5n + something
What number would match up with the 0 term?
The rule must be -5n+12
5(1)  c  7
5  c  7
c  12
Example
You can find the 20th term by replacing n with 20
in the function rule: -5n+12
-5n+12
5n  12 
5(20)  12 
100  12 
8 8
0
12
Rules that generate a sequence
with a constant difference are linear
functions. This can be seen by
graphing (term number, value).
Notice the line is y = -5x + 12
Example
• If you placed 200 points on a line, into how many
non-overlapping rays and segments does it divide
the line?
You need to find a rule that relates the number
of points placed on a line to the number of parts
created by those points.
Sketch one point dividing a line.
One point divides a line into 2 rays.
Example
• Continue collecting data
Sketch two points dividing a line.
Two points divides a line into 2 rays and 1
segment.
Example
• Continue collecting data
Sketch three points dividing a line.
Three points divides a line into 2 rays and 2
segment.
Let’s start collecting the data we have gathered to
see if we see a pattern.
0
2
-1
1
Complete the values for 4, 5 and 6 points on a line.
Look for any patterns and try to write a function rule
for each line.
Answer the question for 200 points.
1
2
3
Each year the tree grows large. The
numbers below the tree is the age of the
tree. How many small branches will a ten
year old tree have? Explain your
reasoning.
How many small branches would a 20
year old tree have?
0
-3
0
-2
0
-12
6n - 3
117
-3n + 4
-56
8n-12
148