Lesson 4.12

Explicit Formulas

Don't take the stairs one by one. Take the elevator straight to the top floor. Calculating any term instantly.

Introduction

Recursive formulas are great if you just need the next step. But if you want the 100th step, you don't want to calculate 99 others first. You need an Explicit Formula—a rule that lets you jump directly to any term number ().

Past Knowledge

Lesson 4.11 (Recursive Formula). We still need and .

Today's Goal

Use the formula to find any term.

Future Success

This is exactly the same as . We just write it differently.

Key Concepts

Why (n - 1)?

Imagine taking steps.

Start
You are standing at . You haven't taken any steps yet.
Step 1
To get to , you take 1 step of size d. ()
Step 2
To get to , you take 2 steps. ()
Step 99
To get to , you take 99 steps. ()

The Teleporter Formula

"Start at term 1, and add the difference (n-1) times."

Worked Examples

Example 1: Find the Formula

Basic

Find the explicit formula for:

Identify Parts

(First number)

(Adding 3)

Plug In

Ideally, simplify it: .

Example 2: Find the 50th Term

Intermediate

Use the explicit formula from Example 1 () to find .

Substitute n=50

We are looking for the 50th spot in line.

Calculate

a_50 = 151

Example 3: Decreasing Sequence

Advanced

Find the explicit formula for and then find .

Step 1: Identify Parts

(First term)

(Decreasing — watch the negative!)

Step 2: Write & Simplify

Distribute:

Step 3: Find

Common Pitfalls

Putting in 'd' Instead of 'n'

If asked for the 10th term, . Sometimes students plug 10 into the spot by mistake. Remember stands for "Number" (position number).

Distributing Errors

When simplifying , remember to distribute the 3 to BOTH terms inside. . Don't forget the negative sign.

Real-Life Applications

Savings Accounts: Simple interest is an arithmetic sequence. If you start with $100 () and earn $5/year (), how much will you have in 30 years ()? You use the explicit formula: . Recursive would take forever.

Practice Quiz

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