Explicit and Recursive Function Forms

Lesson 1 of 2: Direct Formulas and Step-by-Step Builds

In this lesson:

• Write explicit formulas — compute any output directly
• Write recursive definitions — build from the previous value
• Verify both forms generate the same sequence

Learning Objectives

By the end of this lesson, you will be able to:

1. Write an explicit expression giving directly in terms of
2. Write a recursive process defining in terms of
3. Translate between explicit and recursive forms for linear and exponential functions
4. Verify that both representations generate the same sequence of values
5. Explain the trade-off: explicit gives direct access; recursive builds step by step
6. Choose the appropriate form based on the question being asked

How Do You Find the 100th Term?

You know the sequence starts: 3, 7, 11, 15, 19, ...

• Each term is 4 more than the previous
• To find the 100th term... do you need terms 1 through 99 first?
• Or is there a smarter way?

Think about which approach you'd actually use.

Explicit Expressions: Direct Access

An explicit expression gives directly in terms of .

• Compute any output without knowing previous outputs
• Examples: , ,
• Want the 100th term? Substitute . Done.

Explicit = direct access to any output

Example: Plumber Charges (Linear)

Scenario: A plumber charges $75 plus $40 per hour.

• Cost for 6 hours:
• No need to compute through first

Example: Investment Growth (Exponential)

Scenario: An investment of $1000 earns 6% per year.

• Value after 20 years:
• Direct substitution — no intermediate values needed

The power of explicit: skip straight to any value you need.

Quick Check

A gym membership costs $25 to join plus $15 per month.

Which explicit formula models total cost after months?

• A)
• B)
• C)

Think before the next slide...

The Power of Explicit Form

With an explicit formula, you have direct access to any output:

• Find in one step — no intermediate values
• Useful when the question asks for a specific, distant term
• Works best for: linear, exponential, polynomial, and many other types

When you can find it, explicit form is the preferred representation.

From Explicit to Recursive

Some processes are naturally step-dependent:

• Bank balance depends on last month's balance plus interest
• Population depends on last year's count plus new births
• Each new value is built from the previous one

Recursive processes capture this dependence directly.

Recursive Processes: Building from Previous Values

A recursive process has two required parts:

1. Initial value: $f(0) = $ some starting number
2. Recursion rule: $f(n) = $ [expression involving ]

You must compute sequentially:

Both parts are required — the rule alone is incomplete.

Plumber — Recursive Form

Same scenario, different representation:

• Initial: (base fee, zero hours)
• Rule: (each hour adds $40)

Build the sequence:

Investment — Recursive Form

Same investment, recursive representation:

• Initial:
• Rule:

Build the first few values:

To find : must compute through first.

A Naturally Recursive Scenario

Rabbit population with mortality:

"Each generation produces twice as many offspring, but 10 don't survive."

Writing an explicit formula for this is hard — the recursion captures the process directly.

Computing Recursive Terms

Practice: ,

Key: means the previous OUTPUT — not .

Quick Check

For , , compute through .

Build each term from the previous one — no skipping.

Guided Practice: Write the Recursive Form

Scenario: An account holds $500. Each year the balance is multiplied by 1.04.

Write the recursive definition:

• Initial: $V(0) = $ ___
• Rule: $V(t) = $ ___ ___

What is the initial value? What operation happens each step?

The Recursive Trade-off

Neither is "better" — each has a purpose.

Practice Problems

Compute 5 terms and write both forms for each:

1. "A tree is 4 feet tall and grows 2.5 feet per year."
2. "A bank account starts at $800 and earns 3% interest annually."
3. Given: , — write the explicit form.

Work through each, then advance for answers.

Answers

1. Tree: ; Recursive: ,

2. Bank: ; Recursive: ,

3. ✓ — verify: ✓, ✓

Both forms define the same function — just different packaging.

Key Takeaways

✓ Explicit expressions give direct access — compute any output in one step

✓ Recursive processes build step by step — initial value AND rule required

✓ Both forms define the same function — explicit for distant terms, recursive for step-dependent processes

Watch out: To find recursively, you must compute through — no shortcut

Watch out: The rule alone is incomplete — generates infinitely many sequences without a starting value

Watch out: means the previous OUTPUT, not

What's Next

Lesson 2 of 2: Procedural Descriptions and Translation Between Forms

• Procedures: when neither formula nor simple recursion fits
• Translation: convert explicit recursive for linear and exponential
• Verification: prove both forms give the same values
• Choosing: which form is best for which question?

Continue to Lesson 2 when ready.