section 2-2 finding the nth term. you have been looking at different sequences each time you were...

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SECTION 2-2 Finding the nth Term

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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?

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Page 1: 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,

SECTION 2-2Finding the nth Term

Page 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,

• 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, _?_, _?_

Page 3: 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,

• 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?

Page 4: 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,

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.

Page 5: 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,

1 2 3 4 5 6 7 8

What will the 12th term be?

What will the 10th term be?

3, 5, 7, 9, 11, _?_, _?_ 1 2 3 4 5

Page 6: 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,

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.

Page 7: 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,

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

Page 8: 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,

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.

Function Rule

3

6

5

10

8

16

1/2

1

-3

-6

n

2n

Rule is to double the input number

Page 9: 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,

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.

Page 10: 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,
Page 11: 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,

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 ______. •If a sequence had a difference -2, then the number in from of the n (the coefficient of n) is ______.•If a sequence had a difference 3, then the number in from of the n (the coefficient of n) is ______.

4

-2

3

Page 12: 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,

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?

Page 13: 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,

• 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

Page 14: 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,

0

5

0

-2

07

Page 15: 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,

0

-5

Page 16: 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,

0

-3

Page 17: 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,

0

5

Page 18: 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,

0

-2

Page 19: 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,

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?

013

7n + 13Will this expression generate the other terms?

Page 20: 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,

Example• Find the rule for the sequence 7, 2, -3, -8, -13, -18

The rule is decrease by 5, so the rule must be -5n + something

5 1 75 712

( ) cc

c

The rule must be -5n+12

012

What number would match up with the 0 term?

Page 21: 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,

ExampleYou can find the 20th term by replacing n with 20 in the function rule: -5n+12

5 125 20 12

8100 128

( )n

-5n+12

Page 22: 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,

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

012

Page 23: 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,

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.

Page 24: 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,

Example• Continue collecting data

Sketch two points dividing a line.

Two points divides a line into 2 rays and 1 segment.

Page 25: 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,

Example• Continue collecting data

Sketch three points dividing a line.

Three points divides a line into 2 rays and 2 segment.

Page 26: 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,

Let’s start collecting the data we have gathered to see if we see a pattern.

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.

02-11

Page 27: 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,

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?

Page 28: 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,

-20

0

-30

-12

6n - 3 117

-3n + 4 -56

8n-12 148

Page 29: 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,
Page 30: 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,
Page 31: 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,
Page 32: 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,