Given the sequence f(n) below.
Ɛhia
1
Ɛhia
1
Ɛhia
1
Describe the pattern used to get from one term to the next.
Describe the pattern used to get from one term to the next.
OBJECTIVES & STANDARDS
Math Objectives
Write recursive formula for a given sequence
Use a recursive formula to model debt repayment
Common Core Math Standards
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
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
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!
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.
Patrice says she used the solution for 3 discs to help her solve the puzzle for 4 discs. How could she do that?
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.
Describing Sequences
Let’s practice describing recursive sequences and finding patterns in them.
Describe the pattern used to get from one term to the next.
Given the sequence f(n) below.
Describe the pattern used to get from one term to the next.
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…
The Tower of Hanoi
Let’s write a recursive formula for the Tower of Hanoi problem in the Intro.
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.
Check your formula by using f(4) to find f(5). Show your work.
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.
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?
Describe the pattern for the number of moves as required the number of disks increases. If you prefer, you can draw a representation instead.
