Recursively Defined Sequences Party Tables: Part 2 Recursively Defined Sequences

= party guest = party table

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

Working Time

Record the following new information in your math Cheat Sheet

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, …

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

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: ? = ? “?”

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

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

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

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

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

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

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

Concert Hall Seats

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

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

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

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

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