1. Recursively Defined Sequences
Party Tables: Part 2
Recursively Defined Sequences

2. = party guest
= party table

3. _____ = number of party tables _____ = number of party guests
Diagram #1
_____ = number of party tables
_____ = number of party guests

4. _____ = number of party tables _____ = number of party guests
Diagram #2
_____ = number of party tables
_____ = number of party guests

5. _____ = number of party tables _____ = number of party guests
Diagram #3
_____ = number of party tables
_____ = number of party guests

6. 1. How many people can sit at 10 tables pushed together?
2. How many tables are needed to sit 32 people?

7. Let’s organize our data.
Number of Tables
Number of Guests

8. Let’s organize our data.
Number of Tables
Number of Guests 1 4

9. Let’s organize our data.
Number of Tables
Number of Guests 1 4 2 6

10. Let’s organize our data.
Number of Tables
Number of Guests 1 4 2 6 3 8

11. Let’s organize our data.
Number of Tables
Number of Guests 1 4 2 6 3 8 •
Continue
the sequence in your table until you have 32 people.

12. Let’s organize our data.
Number of Tables
Number of Guests 1 4 2 6 3 8 • 15 32
Continue
the sequence in your table until you have 32 people.

13. 3. Describe in words any patterns that you see from this situation.

16. Recursive Sequences 1 4 2 6 3 8 10 5 12 14 7 16 Number of Tables
Number of Guests 1 4 2 6 3 8 10 5 12 14 7 16

17. GET OUT YOUR CHEAT SHEET (PASS OUT RULE OF FOUR)

18. each step of a pattern is dependent on the step that comes before it.
Recursive Sequences
Recursion
is a process
in which
each step of a pattern is dependent on the step that comes before it.
Number of Tables
Number of Guests 1 4 2 6 3 8 10 5 12 14 7 16

20. Recursive Sequences Record this data table on your handout. 1 4 2 6 3
Number of Tables
Number of Guests 1 4 2 6 3 8 10 5 12 14 7 16

21. Describe in words the pattern that you see from this situation.
Recursive Sequences
Describe in words the pattern
that you see from this situation.

22. Recursive Sequences Graph the data
Number of Tables
Number of Guests 1 4 2 6 3 8 10 5 12 14 7 16
Graph
the
data
Number of Guests
Number of Tables

23. Working
Time

25. Record the following new information in your math Cheat Sheet

26. Recursive Sequences
A sequence is an ordered set of numbers.
Number of Guests 4 6 8 10 12 14 16
The pattern formed by the number of guests can be represented by a sequence:
4, 6, 8, 10, 12, …

27. Developing a Recursive Formula
4, 6, 8, 10, 12, …
A recursive sequence can be described efficiently with mathematical symbols, using
a recursive formula.

28. Developing a Recursive Formula
4, 6, 8, 10, 12, …
Each number in the sequence is called a term.
1st term: u1 = 4
“u sub one equals four”
2nd term: u2 = 6
“u sub two equals six”
3rd term: u3 = ?
“?”
4th term: ? = ?
“?”

29. Developing a Recursive Formula
4, 6, 8, 10, 12, …
This is a recursive sequence as each term depends on the previous term.
1st term: u1 = 4
u1 = 4 initial value
2nd term: u2 = 6
= 4 + 2
= u1 + 2
3rd term: u3 = 8
= 6 + 2
= u2 + 2
4th term: u4 = 10
= 8 + 2
= u3 + 2

30. Developing a Recursive Formula
1st term: u1 = 4 initial value
2nd term: u2 = 6
= 4 + 2
= u1 + 2
3rd term: u3 = 8
= 6 + 2
= u2 + 2
4th term: u4 = 10
= 8 + 2
= u3 + 2
For this sequence,
any term =
previous term + 2 un
= un–1 + 2

31. Developing a Recursive Formula
This recursive formula has three parts:
The first term:
u1 = 4
The general term:
un =
un–1 + 2
where n ≥ 2

33. Developing a Recursive Formula
This recursive formula has three parts:
The first term:
u1 = 4
The general term:
un =
un–1 + 2
where n ≥ 2

34. u1 = un = un–1 + d Arithmetic Sequence the first term
The general term (recursive rule):
un =
un–1 + d
d = the common difference from term to term

35. d = the common difference
Arithmetic Sequence
4, 6, 8, 10, 12, …
d = the common difference
d = any term – previous term
d = 6 – 4 = 2
d = 10 – 8 = 2
d = 8 – 6 = 2
d = 12 – 10 = 2

36. u1 = un = un–1 + d Arithmetic Sequence the first term
The general term (recursive rule):
un =
un–1 + d
d = the common difference from term to term

37. Concert Hall Seats

38. Concert Hall Seats etc. Row 1: 59 seats Row 2: 63 seats

39. 59, 63, 67, … un = ? Concert Hall Seats Arithmetic Sequence: u1 = ?
Recursive Formula:
u1 = ?
d = ?
un = ?
Un-1 + 4

40. u1 = ? un = ? Practice 1. The first term is 40. Keep adding 11.
Write the first five terms of the sequence.
Write the recursive formula.
1. The first term is 40. Keep adding 11.
u1 = ?
un = ?
Un-1 +11 40

41. u1 = ? un = ? Practice 2. Start at 27. Keep subtracting 8. 27 Un-1 − 8
Write the first five terms of the sequence.
Write the recursive formula.
2. Start at 27. Keep subtracting 8.
u1 = ?
un = ? 27
Un-1 − 8

42. u1 = ? un = ? Practice 3. ___ , __ , 42, ___ , 50, ... 34 38 46 34
Write the first five terms of the sequence.
Write the recursive formula. 34 38 46
3. ___ , __ , 42, ___ , 50, ...
u1 = ?
un = ? 34
Un-1 + 𝟒