Lesson 4.11

Recursive Formulas

To know where you're going, you have to know where you've been. Defining a sequence one step at a time.

Introduction

A Recursive Formula is like a set of instructions for a scavenger hunt. It tells you two things: where to start, and how to get to the next spot from your current spot.

Past Knowledge

Lesson 4.10 (Arithmetic Intro). We need the Common Difference () here.

Today's Goal

Write a recursive formula using the notation .

Future Success

Recursive thinking is the foundation of computer loops (for-loops and while-loops).

Key Concepts

Translating the Symbols

The Symbol

"Current Term"

Also called "Toady" or "Next"

The Symbol

"Previous Term"

Also called "Yesterday" or "Before"

The Formula Recipe

"To get Today's number, take Yesterday's number and add d."

(Don't forget to state where you start! )

Worked Examples

Example 1: Writing the Formula

Basic

Write a recursive formula for:

Step 1: Find Start ($a_1$)

The list starts at 7.

Step 2: Find Difference ()

. We add 3 every time.

Step 3: Combine


Example 2: Using the Formula

Intermediate

Given and , find the first 4 terms.

Just follow the instructions

Start at 20.

Next = Previous - 5.

  • Term 1: 20
  • Term 2:
  • Term 3:
  • Term 4:
20, 15, 10, 5

Example 3: Word Problem

Advanced

Maya has $50 in her savings account and deposits $8 every week. Write a recursive formula for the balance after weeks, then find the balance after 5 weeks.

Step 1: Identify the Parts

Starting balance:

Amount added each week:

Step 2: Write the Formula


Step 3: Generate Terms

  • Week 1: 50
  • Week 2:
  • Week 3:
  • Week 4:
  • Week 5:
After 5 weeks: $82

Common Pitfalls

Forgetting the Start

Simply writing is not enough! That just says "add 3 forever". But start where? 0? 100? You MUST include the line.

Only Easy for Small N

Recursive is great for finding the "next" term. It is TERRIBLE for finding the 100th term. You would have to calculate all 99 steps before it.

Real-Life Applications

Spreadsheets: Excel works recursively. You type =A1+5 in cell A2, then drag it down. Each cell looks at the one above it and adds 5. You are literally coding a recursive sequence.

Practice Quiz

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