Write Arithmetic and Geometric Sequences

In this lesson:

• Write sequences two ways: step-by-step and jump-to-any-term
• Tell which real situations need which kind

What You'll Be Able to Do

By the end, you should be able to:

1. Write arithmetic and geometric sequences, step-by-step and by formula
2. Find or from a list of terms
3. Translate between the two forms
4. Model real situations with sequences
5. Pick the efficient form for a term

Two Jobs, Two Kinds of Change

Job offer: $35,000 to start, $2,000 raise every year
Car purchase: $30,000 now, loses 15% of its value every year

What's the salary at year 10? What's the car worth at year 5?

Building the Salary, Year by Year

Keep adding $2,000. What's year 5? Year 10 — did you add nine times or ten?

Meet the Common Difference, d

• Each year adds the same amount: $2,000
• That fixed amount is the common difference,
• A sequence with a constant is an arithmetic sequence

Example: 5, 8, 11, 14, 17 — here

Recursive Form: Step by Step

"Take the previous term, add $2,000." Each new term needs the one before it.

Why (n − 1), Not n

• From year 1 to year 10, you add only 9 times
• From term 1 to term , you add exactly times
• Using times instead of shifts every term wrong

Explicit Form: Jump to Any Term

For the salary:

Keep the visible — don't simplify yet.
Check: ✓

Simplifying — a Separate Step

Year 10: — that's $53,000.

Watch out: the simplified form has , not — that's normal algebra, not the off-by-one mistake

The Common Difference Is the Slope

• looks like
• Common difference is the slope
• First term lines up with the -intercept idea

Quick Check: Find the Difference

Find the common difference and write both forms:

Think about it — the difference doesn't have to be positive.

Now the Car: Multiply, Don't Add

Each year, multiply by 0.85. Predict — don't compute yet — what year 5 will be.

Meet the Common Ratio, r

• Each year multiplies by the same factor: 0.85
• That fixed factor is the common ratio,
• A sequence with a constant is a geometric sequence

Example: 3, 6, 12, 24, 48 — here

Recursive Form: Multiply Each Step

"Take the previous term, multiply by 0.85." Same recursive idea, different operation.

Explicit Form: Unrolling the Multiplication

Same step-count logic: reaching term takes multiplications by

Checking the Formula and Resolving Year 5

For the car:

Check: ✓

Resolving the prediction — year 5:

That's about $13,311.

The Common Ratio Is the Base

• looks like
• Common ratio is the base
• First term is the initial value

Geometric sequences are exponential functions on whole-number inputs.

Growth, Decay — Same Formula Either Way

One formula, two behaviors — the sign of " vs. 1" decides which.

Predict First: Can r Be Negative?

Does a common ratio have to be positive?

• A. Yes — ratios describe growth or shrinkage, always positive
• B. No — a negative ratio is still valid

Pick A or B before advancing. There's a reason this trips people up.

Yes — Negative Ratios Are Valid

• Common ratio:
• Signs alternate because a negative times a negative flips back positive

Quick Check: Growth or Decay?

Find the common ratio and classify:

Is this a growing sequence or a shrinking one — and by what factor?

Could You Build the Explicit Form Yourself?

Arithmetic:
Geometric:

If I only gave you the recursive rule, could you have written these?

Translating Arithmetic: Recursive to Explicit

Given:

Find: ,

Check: ✓

Translating Geometric: Recursive to Explicit

Given:

Find: ,

Check: ✓ — , and ✓

Your Turn: Work the Reverse Direction

Given the explicit form:

• Is this arithmetic or geometric? _______
• What is ? _______
• Write the recursive form: $a_1 = 200, \quad a_n = $ _______

Find the Error in This Formula

Someone wrote this explicit formula. Something's wrong.

Check it: what does this formula give for ? Should it give 4?

The Fix: r to the (n − 1)

Check: ✓

If you use instead: — wrong, unless

Quick Check: Translate in Reverse

Given:

• Find (hint: rewrite as )
• Write the recursive form
• Verify with two terms

A Tank Fills at 50 Gallons a Minute

Before we formalize modeling — pick a form: arithmetic, or geometric?

The Rule for Classifying Real-World Change

Fixed amount each step → arithmetic. Fixed percent/factor each step → geometric.

Watch Out: d or r, Which One?

A sequence: — someone computes anyway:

Not constant — dividing an arithmetic sequence never gives a steady ratio

Not Every Sequence Fits a Box

Differences: 2, 3, 4, 5 — not constant. Ratios: 3, 2, 5/3, 3/2 — not constant.

Complete Model: A Fixed Monthly Deposit

"You deposit $100 per month, no interest."

• Type: Arithmetic — fixed amount added each month
• After months:

Complete Model: A Compounding Deposit

"You deposit $1,000 once and earn 0.5% monthly interest."

• Type: Geometric — fixed percent multiplied each month
• After months:

What If the Car's Rate Changes?

Depreciates $3,000/year instead of 15%/year — same starting price.

• Type: geometric → arithmetic ()
• Year 5, flat:
• Year 5, compounding:

Solo Commit: Arithmetic or Geometric?

"A phone loses 10% of its battery charge every hour it's left on." Write your answer — no discussion yet.

Your Turn: Full Model, No Scaffolding

"You start a savings plan by depositing $50 every month, no interest."

Classify the type. Find and or . Write both the recursive and explicit forms. Then answer: how much is saved after 12 months?

Two Formulas, the Same Mistake to Watch For

Arithmetic: Wrong:   Right:

Geometric: Wrong:   Right:

Both come from miscounting the steps from term 1 to term

What This Lesson Really Taught You

Fixed amount or fixed factor tells you the family — recursive and explicit are just two views of the same or .

Next: a bare table, no "raise" or "percent" to hint at it. Can you name the family and build the function? That's HSF.LE.A.1–2.