Lesson 4.13

Converting Recursive to Explicit

Two languages, one meaning. Whether you give step-by-step directions or a teleporter address, you end up at the same number.

Introduction

We have learned two ways to describe a sequence. Recursive (Next = Previous + d) and Explicit (). Both rely on the same two ingredients: the Start () and the Difference (). If you can find those, you can translate between them.

Past Knowledge

Lessons 4.11 and 4.12. You must know what the formulas look like.

Today's Goal

Extract and from one formula and plug them into the other.

Future Success

Being able to switch forms is crucial for standardized tests (SAT/ACT) which often ask for "the equivalent form".

Key Concepts

The "Parts" Strategy

Don't try to memorize a "conversion formula". Just find the ingredients.

Ingredient 1: Start

a₁

Recursive: Often listed separately line ()

Explicit: The number sitting alone by itself.

Ingredient 2: Difference

Recursive: The number added to .

Explicit: The number multiplied by .

Once you have and , you can build ANYTHING.

Worked Examples

Example 1: Recursive → Explicit

Basic

Convert:

Identify Parts

(Given directly)

(Because we add 5)

Build Explicit

Use

Example 2: Explicit → Recursive

Intermediate

Convert:

Identify Parts

(The lonely number)

(The multiplier)

Build Recursive

Example 3: From Simplified Explicit

Advanced

Convert to Recursive.

Warning: Simplification Hides

We can't just see easily. We have to calculate it.

Plug in : . So Start = 10.

Find d

The number attached to is always the Difference (Slope). So d = 3.

Build Recursive

Common Pitfalls

Simplified Forms

Students see and assume . No! That 7 is the "zero term" (y-intercept). Always plug in to be safe.

Forgetting the Start (Again)

Even when converting TO recursive, you MUST state . A recursive formula is useless without a starting point.

Real-Life Applications

Computer Science: Programmers constantly convert between Iteration (loops/recursive thinking) and Direct Calculation (explicit formulas). Explicit is faster ( constant time), but Recursive is often easier to write logic for.