The Sequence Race

Topics 13.1, 13.2, 13.3, 13.4, 13.5 -- Sequences, Recursive Sequences, Arithmetic, Geometric, Applications

Two job offers. One starts higher and gives you a flat $2,000 raise every year. The other starts lower and gives you 6% every year. The first one pays more for over a decade. Take it anyway and you will lose a lot of money. Here is exactly how much.

Discover: The Salary Race

Step 1 of 8

Company A offers $40,000 to start with a $2,000 raise every year. Company B offers $35,000 to start with a 6% raise every year. Which offer is better?

Step 2 of 8

Watch the salaries grow year by year. Use the button to advance one year at a time and see who is winning.

Company A (arithmetic) Company B (geometric)

Step 3 of 8

Does Company B ever catch up? If so, predict the year it happens.

B overtakes A at year:

Step 4 of 8

Now advance to year 25. See how far apart the salaries have become.

Step 5 of 8

Name the pattern. Company A adds the same amount each year. Company B multiplies by the same amount each year. What are these types of sequences called?

Company A's pattern (adding $2,000 each year):

Company B's pattern (multiplying by 1.06 each year):

Step 6 of 8

Two ways to describe the next term. Recursive: "take the previous year and add $2,000." Explicit: "start at $40,000 and add $2,000 for each year after the first." The recursive form needs every prior year. The explicit form jumps straight to any year.

What is Company A's salary in year 40 using the explicit formula? (Within $100)

Year 40 salary ($):

Step 7 of 8

The general forms. For an arithmetic sequence, each term adds a common difference $d$. For a geometric sequence, each term multiplies by a common ratio $r$.

Why is it $n - 1$ instead of $n$ in both formulas?

Step 8 of 8

Experiment with different values of $d$ (the flat raise) and $r$ (the percentage multiplier). Can you find settings where the arithmetic sequence wins forever?

Common difference $d$: $2,000

Common ratio $r$: 1.06

Explore: Build Your Own Sequence Race

Set the first term, common difference, and common ratio. Both sequences update live.

First term $a_1$: $40,000

Common difference $d$: $2,000

Geometric first term: $35,000

Common ratio $r$: 1.06

Show terms up to $n$: 25

Logarithmic vertical scale

Arithmetic: $a_n = a_1 + (n-1)d$ Geometric: $b_n = b_1 \cdot r^{n-1}$

Adjust the sliders to see how the first term, common difference, and common ratio change the race. Toggle the log scale to see why geometric growth looks like a straight line on a logarithmic axis.

Try This

Starter: Jar vs. Investment

You put $200 a month in a jar (no interest). Your friend invests $200 a month at 0.5% monthly return compounded monthly.

After $n$ months, your jar total is $200n$ (arithmetic). Your friend's total uses the geometric series formula.

At 5 years (60 months), your jar has $12,000. What does your friend have? (Within $100)

Friend's total ($):

When does the gap first exceed $10,000? (Within 12 months)

Month:

Stretch: Video vs. Article

A video gets 500 views on day 1 and triples daily (geometric, $r = 3$). An article gets 10,000 views on day 1 and adds 500 daily (arithmetic, $d = 500$).

On day 1, the article is 20 times bigger. The video starts tiny but triples every day.

On what day does the video pass the article in daily views? (Within 1 day)

Day:

What does this say about why platforms chase early velocity in content?

Challenge: Bouncing Ball

A ball drops from 20 feet and rebounds to 75% of its previous height each bounce.

This is geometric with $a_1 = 20$ feet and $r = 0.75$. List the first 8 bounce heights.

What is the height after bounce 8? (Within 0.1 feet)

Height (ft):

Is this sequence arithmetic or geometric?

Does the ball ever stop bouncing? Does it travel a finite total distance? (Set this up -- Simulation 30 finishes it.)

Distinguish arithmetic sequences (constant difference) from geometric sequences (constant ratio)
Write both recursive and explicit formulas for arithmetic and geometric sequences
Explain why the explicit formula uses $n - 1$
Compare the long-term behavior of linear vs. exponential growth
Apply sequences to real-world scenarios involving salaries, savings, and growth

Quick Check

A sequence starts at 5 and each term is multiplied by 3. What is the 6th term?

Instructor Notes

Teaching Notes

The salary comparison is the most effective hook for this topic because students immediately see the stakes. The "arithmetic wins for 9 years" setup is designed to produce genuine surprise -- most students will pick Company A after seeing it win year after year. That surprise is the entire pedagogical strategy: a wrong prediction based on real evidence, followed by the explanation of why exponential growth always wins eventually.

The recursive-to-explicit transition (Step 6) is where students struggle most. Emphasize that recursive is natural ("add $2,000 to last year") but impractical for distant terms. The explicit formula is the shortcut that lets you skip ahead.

Common Student Errors

Using $n$ instead of $n - 1$ in explicit formulas. Reinforce: "the first term has had zero raises."
Believing a bigger starting value or bigger common difference can beat geometric growth permanently. The crossover always happens -- it may take longer, but it always comes.
Confusing the common ratio with the growth rate. $r = 1.06$ means 6% growth, not 106% growth.
Treating the recursive formula as if it lets you jump to any term. It requires all prior terms.

Discussion Questions

If you plan to work somewhere for only 5 years, does the geometric offer still win? When does the time horizon change the decision?
What real-world quantities grow arithmetically? What grows geometrically? Which is more common in nature and finance?
Could you design a raise structure that beats both arithmetic and geometric growth? What would it look like?

Exam Connection

Expect problems asking students to identify sequences as arithmetic or geometric, write explicit and recursive formulas, and find specific terms. The $n - 1$ exponent is a frequent error on exams. Application problems involving salary comparisons, loan payments, or depreciation are common. Students should be comfortable with both forms and know when to use which.