Sequences Are Functions

In this lesson:

• Recognize sequences as functions with integer domains
• Use function notation and graph sequences as discrete points
• Write explicit formulas and recursive definitions

What You Will Learn Today

1. Explain why a sequence is a function with integer domain
2. Use both and notation for sequence terms
3. Write and evaluate explicit formulas for sequences
4. Write recursive definitions with initial term and recurrence relation
5. Graph sequences as discrete points and explain why

What Is the Next Term in This Pattern?

• Most people say 23 — add 4 each time
• But how do you know? You recognized a rule connecting position to output

Can we describe this pattern using the language of functions?

A Sequence IS a Function

• One input → one output: satisfies the function definition
• Domain: ; Range:

Reading a Sequence as a Function Table

• Each input (position) maps to exactly one output (term)
• Domain: a subset of the integers —
• A sequence is a function — apply every function concept to it

Two Notations — Same Meaning

Both mean "the term at position ":

• and mean the same thing
• Both are function notation — input is position, output is the term
• You will see both in textbooks — they are interchangeable

Graphing a Sequence as Discrete Points

Why Don't We Connect the Dots?

• No term at — connecting dots claims otherwise
• Same formula, different domain, different graph

Transition: Two Ways to Define a Sequence

Both forms describe the same function.

Explicit Formulas — Direct Access

An explicit formula gives directly in terms of .

• ✓, ✓
• — no build-up needed

How to Derive an Explicit Formula

1. Common difference:
2. General form:
3. Simplify:
4. Verify: ✓, ✓, ✓

Check: Write the Explicit Formula

Work through these steps:

1. Find the common difference
2. Write the general form
3. Simplify and verify

Your answer: $a(n) = $ ?

(Answer on the next slide)

Explicit Formula: Check Your Answer

• Common difference: , so
• General form:
• Verify: ✓, ✓

Recursive Definitions — Step by Step

Two required parts:

1. Initial condition — the starting term:
2. Recurrence relation — how to get the next:

Building the sequence:

Why the Initial Condition Is Required

• All three sequences satisfy
• Each has a different starting point, producing different terms
• Without , the recurrence alone is ambiguous

Rule: A recursive definition is incomplete without initial condition(s).

Geometric Sequences — Both Forms

— ratio

Explicit:

• ✓, ✓

Recursive: , for

Same function — two representations, same input-output pairs.

Check: Write Explicit and Recursive Forms

1. Identify the type: arithmetic or geometric?
2. Write the explicit formula $a(n) = $ ?
3. Write the recursive definition

(Answer on the next slide)

Both Forms: Explicit and Recursive Answers

• Type: Geometric —
• Explicit: — verify: ✓, ✓
• Recursive: , for

Independent Practice: Three Sequence Problems

Work independently. Show your steps.

1. Write as a function table with domain . State the range.

2. For the sequence , , find .

3. Write the explicit formula for

Answers: Check Your Practice Work

1. — Range:

2. , , ,

3. Ratio →

• Verify: ✓, ✓

Key Takeaways: Sequences and Their Definitions

• A sequence is a function — integer domain, isolated-dot graph
• Explicit: substitute directly — instant access to any term
• Recursive: initial condition + recurrence — both required
• Both forms describe the same function

Watch out: No connected dots. No missing initial conditions.

Coming Up: Arithmetic, Geometric, and Fibonacci

In Lesson 2:

• Arithmetic — constant difference, the discrete analog of linear functions
• Geometric — constant ratio, the discrete analog of exponential functions
• Fibonacci — a two-term recursive definition from the standard
• Classification drill: arithmetic, geometric, or neither?