Understanding Sequential Patterns with Functions and Rules

1. Section 2-2 Finding the nth Term

2. 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, _?_, _?_

3. 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?

4. 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.

5. 1 2 3 4 5 6 7 8 1 2 3 4 5 3, 5, 7, 9, 11, _?_, _?_ What will the 12th term be? What will the 10th term be?

6. 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.

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

8. 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 8 n 1/2 3 -3 Function Rule Rule is to double the input number 16 -6 1 2n 10 6

9. 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.

11. 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

12. 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?

13. 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

14. 0 5 0 -2 0 7

15. 0 -5

16. 0 -3

17. 0 5

18. 0 -2

19. 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?

20. 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

21. Example You can find the 20th term by replacing n with 20 in the function rule: -5n+12 -5n+12

22. 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

23. 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.

24. Example • Continue collecting data Sketch two points dividing a line. Two points divides a line into 2 rays and 1 segment.

25. Example • Continue collecting data Sketch three points dividing a line. Three points divides a line into 2 rays and 2 segment.

26. 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.

27. 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?

28. 0 -3 6n - 3 117 0 -2 -56 -3n + 4 0 148 -12 8n-12