Arithmetic, Geometric, and Fibonacci

In this lesson:

• Classify sequences as arithmetic, geometric, or neither
• Write explicit and recursive formulas for each type
• Compute Fibonacci terms and explain why two initial values are needed

What You Will Learn Today

1. Write explicit and recursive formulas for arithmetic sequences
2. Write explicit and recursive formulas for geometric sequences
3. Classify a sequence as arithmetic, geometric, or neither
4. Compute Fibonacci terms and explain why two initial terms are needed
5. Connect arithmetic sequences to linear functions graphically

Quick Review from Lesson 1

is a function with — but what type?

• Differences: — constant → arithmetic
• Ratios: — not constant → not geometric

Today we name and formalize both types.

Arithmetic Sequences — Definition and Formulas

An arithmetic sequence has a constant difference between consecutive terms.

General formulas:

• Explicit:
• Recursive: , for

Think: Arithmetic → Add (constant addition each step)

Arithmetic Example: Find Both Formulas

Explicit: , so

• Verify: ✓, ✓

Recursive: , for

Arithmetic Sequence — Recursive Build

, builds:

Theater: 20 seats in row 1, adding 2 per row.

Arithmetic Graph — Discrete vs. Continuous

• Arithmetic sequence : isolated dots at integers
• Linear function : continuous line
• Both have slope — arithmetic sequences are the discrete analog of linear functions

Check: Classify and Write Formulas

Theater problem: Row 1 has 20 seats, each row adds 2.

1. Is this arithmetic or geometric?
2. Write the explicit formula $s(n) = $ ?
3. Write the recursive definition

(Answer on the next slide)

Geometric Sequences — Definition and Formulas

A geometric sequence has a constant ratio between consecutive terms.

General formulas:

• Explicit:
• Recursive: , for

Think: Geometric → Grow by multiplication

Geometric Example: Find Both Formulas

Sequence:

Explicit formula:

• Ratio:
• Verify: ✓, ✓

Recursive formula:

• , for

Real-World Geometric: Bank Account Growth

$1000 at 5% annual interest:

• Ratio — discrete analog of exponential growth

Check: Classify Three Sequences by Type

Classify each as arithmetic, geometric, or neither. State or :

(Answers on the next slide)

Classification Drill: Identify and Write Formulas

Classify each sequence and write the explicit formula:

(Answers on the next slide)

Classification Drill: Answers and Formulas

1. — arithmetic, ,
2. — geometric, ,
3. — geometric, ,
4. — arithmetic, ,

The Fibonacci Sequence — Two Initial Terms

Computing the First Eleven Fibonacci Terms

Definition: , , for

Each term = sum of two previous terms.

Why Two Initial Terms Are Required

The rule looks back two positions.

Check: Why Two Initial Terms?

The rule with only :

• Can you compute ? (Need — unknown!)
• Can you compute ? (Not given by the rule — undefined!)

Without both and , the Fibonacci sequence cannot begin.

Fibonacci Is Neither Arithmetic nor Geometric

• Differences: — not constant
• Ratios: — not constant

Still a function: integer domain, each input → one output. Only definition is recursive.

Synthesis: All Three Are Functions

• All three are functions with integer domains
• Arithmetic and geometric have explicit formulas; Fibonacci does not
• The number of initial conditions matches how many steps back the rule looks

Synthesis: Three Mixed Practice Problems

Work independently:

1. Compute using , ,

2. Classify — write both formulas

3. A savings account has $500; you add $50 each month. Write . Is it arithmetic or geometric?

Synthesis Practice: Check Your Answers

1.

2. Geometric, : explicit ; recursive ,

3. Arithmetic, :

• ✓, ✓

Key Takeaways: Three Sequence Families

• Arithmetic: constant — discrete analog of linear functions
• Geometric: constant — discrete analog of exponential functions
• Fibonacci: two-term rule — requires two initial conditions

Classify: differences → arithmetic; ratios → geometric; neither → other.
Watch out: All three are functions with integer domains.

What's Next: Building Sequence Formulas

Coming up in HSF.BF.A.2:

• Derive arithmetic and geometric formulas from context
• Model real-world situations as sequence functions
• Connect sequences to linear and exponential function families