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5/10 FA 7.5 Recursive Sequences

Last updated over 2 years ago
14 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
  • Write recursive formula for a given sequence
  • Use a recursive formula to model debt repayment
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.HSF.IF.A.3
  • CCSS.HSF.BF.A.1
  • CCSS.HSF.BF.A.1.A
  • CCSS.HSF.BF.A.1.B
Personal Finance Objectives
  • Calculate how a debt amount decreases over time, given the interest rate and regular payment amounts
National Standards for Personal Financial Education
Managing Credit
  • 1b: Compare the cost of borrowing $1,000 using consumer credit options that differ in rates and fees
OBJECTIVES & STANDARDS
Math Objectives
  • Write recursive formula for a given sequence
  • Use a recursive formula to model debt repayment
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.HSF.IF.A.3
  • CCSS.HSF.BF.A.1
  • CCSS.HSF.BF.A.1.A
  • CCSS.HSF.BF.A.1.B
Personal Finance Objectives
  • Calculate how a debt amount decreases over time, given the interest rate and regular payment amounts
National Standards for Personal Financial Education
Managing Credit
  • 1b: Compare the cost of borrowing $1,000 using consumer credit options that differ in rates and fees
Intro
INTERACTIVE: Tower of Hanoi
The Tower of Hanoi is a mathematical puzzle that has inspired many myths about its origins. According to one version, priests were tasked with solving this puzzle using 64 disks and when the puzzle was completed, the world would end. Explore this classic puzzle to see what patterns you can discover!
Required
1

Complete the puzzle using 3 disks. Try to complete it in the minimum number of moves possible (7). Did you complete it in less than 7 moves?

Required
1

Complete the puzzle using 4 disks, and then 5 disks. Record any observations below, like how many moves you took or what patterns you noticed.

Required
1

Patrice says she used the solution for 3 discs to help her solve the puzzle for 4 discs. How could she do that?

Required
1
The table below shows the minimum number of moves required to solve the puzzle with each number of disks. Complete the last two rows: How many moves would it take to complete the puzzle with 9 disks?_______ With 10 disks?_______

Required
1

Describe the pattern for the number of moves as required the number of disks increases. If you prefer, you can draw a representation instead.

Learn It
Recursive Sequences
As you add more disks to the Tower of Hanoi, you can use the previous solutions that had fewer disks to help you figure out the puzzle. The pattern we discovered for the minimum number of moves is an example of a recursive sequence.
Sequence: a list of numbers
Term: a number in a sequence
Recursive sequence: a sequence in which a term is defined using a previous term.
Example
Here is a sequence: 11, 13, 15, 17, 19, 21…
We could define this sequence recursively by saying it starts with 11 and each term is 2 greater than the previous term.
Required
6
Write a sequence of numbers where the first term is 5 and each term is 2 times the previous term.
_______, _______, _______, _______ ,_______, _______,....
Explore It
Describing Sequences
Let’s practice describing recursive sequences and finding patterns in them.
Required
1
Given the sequence: 20, 10, 5, 2.5, 1.25, 0.625…
What is the 4th Term? _______
Required
1

Describe the pattern used to get from one term to the next.

Given the sequence f(n) below.

Required
1
What is the 3rd term?_______
Required
1
What is the 5th term?_______
Required
1

Describe the pattern used to get from one term to the next.

Learn It
Writing Recursive Formulas
Throughout this lesson, you’ve been noticing patterns and describing how to get from one term to the next.
We can capture that information using a recursive formula.
A recursive formula specifies the first term of the sequence and the steps to get from one term to the next.
The formula includes two parts:
  • One of the terms in the sequence (usually the first)
  • The pattern: how you can find a specific term by using the previous term
  • The notation f(n-1) represents “the previous term”
Example
Here is the recursive formula for the sequence you saw earlier in the Explore It Question 1.
Sequence: 20, 10, 5, 2.5, 1.25, 0.625…

Practice It
The Tower of Hanoi
Let’s write a recursive formula for the Tower of Hanoi problem in the Intro.


Required
1

Write a recursive formula for the sequence of minimum moves in the Tower of Hanoi. You can use this video walkthrough to guide you through the process.

Required
1

Check your formula by using f(4) to find f(5). Show your work.

Required
25

ACTIVITY: DESMOS LINK: MOVE: Matching Recursive Sequences
Let’s practice matching recursive formulas with their sequences. 20 POINTS
This link completes 3 desmos in 1:
1. Card Sort
2. Application Level 1
3. Application Level 2

Write a brief reflection of the desmos activities and I will transfer your Desmos grade here.

The End!