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1.4 — Scott’s Workout
Human review completed .
1. In class: A Solidify Understanding Task
Arithmetic sequences have a constant difference
Scott’s bars show 3, 5, 7, 9 push-ups on days 1–4. Each day, he does 2 more push-ups than the day before. The change from one term to the next is always +2.
To find that change, subtract the earlier term from the next term, keeping the terms in their sequence order:
A sequence with the same change at every step is an arithmetic sequence. That repeated change is called the common difference. “Common” means it is the same for every pair of consecutive terms; “difference” means we find it by subtraction. Scott’s sequence has common difference 2, so we add 2 to get each next term.
The common difference can also be negative. For example, in 11, 9, 7, 5, each term is 2 less than the previous one:
Here the common difference is −2. Keep the negative sign: it tells us the terms decrease. A constant sequence such as 6, 6, 6, 6 has common difference 0.
Recursive rule: start, then add
Let \(S(n)\) be the push-ups Scott does on day n. Day 1 is the first term:
Explicit rule: count the changes
To reach day \(n\) from day 1, there are \(n-1\) increases. Start with 3 and add 2 for each increase:
Worked example — Main 1
Original problem: How many push-ups will Scott do on day 10?
Worked solution: From day 1 to day 10 there are 9 increases, not 10:
Answer: 21 push-ups on day 10. This is not the total over the first ten days.
Main 2–3: The explicit and recursive rules above model every positive whole-number day. The value 1 in \(2n+1\) would be the output at day 0, not the first term. The first term is \(S(1)=3\).
Compare Scott with Aly — Main 4
Aly does 15 push-ups every day. Scott does fewer on days 1–6, the same 15 on day 7, and more on day 8 onward. Therefore Aly is not always doing more; it depends on the day being compared. The rules compare daily amounts.
2. Ready
Work independently first. Use the answers below to check your work after trying the problems, or for help when you are stuck.
Check your work — Ready 1–8
| Problem | First requested value | Second requested value |
|---|---|---|
| 1 | \(f(4)=28\) | \(f(-2)=-2\) |
| 2 | \(f(10)=-19\) | \(f(-1)=3\) |
| 3 | \(f(-5)=-33\) | \(f(0)=-3\) |
| 4 | \(f(9)=-9\) | \(f(-11)=11\) |
| 5 | \(f(2)=25\) | \(f(3)=125\) |
| 6 | \(f(4)=81\) | \(f(1)=3\) |
| 7 | \(f(6)=1000000\) | \(f(0)=1\) |
| 8 | \(f(0)=1\) | \(f(5)=32\) |
3. Set
Work independently first. Use the answers below to check your work after trying the problems, or for help when you are stuck.
Check your work — Set 9–11
| Problem | A — Next terms | B — Difference | C — Recursive rule | D — Explicit rule |
|---|---|---|---|---|
| 9 | 28, 33, 38 | \(5\) | \(f(1)=3\); \(f(n+1)=f(n)+5\) for \(n=1,2,3,\ldots\) | \(f(n)=3+5(n-1)\) |
| 10 | 1, −1, −3 | \(-2\) | \(f(1)=11\); \(f(n+1)=f(n)+(-2)\) for \(n=1,2,3,\ldots\) | \(f(n)=11-2(n-1)\) |
| 11 | −4.5, −6, −7.5 | \(-1.5\) | \(f(1)=3\); \(f(n+1)=f(n)+(-1.5)\) for \(n=1,2,3,\ldots\) | \(f(n)=3-1.5(n-1)\) |
For 10, adding −2 is subtracting 2; for 11, adding −1.5 is subtracting 1.5. In the recursive rule, circle the amount added. In the explicit rule, circle the coefficient of \(n-1\).
4. Go
Work independently first. Use the answers below to check your work after trying the problems, or for help when you are stuck.
Check your work — Go 12–17
| Problem | Answer |
|---|---|
| 12 | 9,000 feet |
| 13 | 6,500 feet |
| 14 | 2,500 feet: 9,000−6,500 |
| 15 | 1,000 feet |
| 16 | 3–4 pm: a drop of 3,500 feet, larger than the other hourly drops. |
| 17 | Elevation; the input is time. |
Additional learning resources
| Resource | What it helps with and scope |
|---|---|
| Khan Academy — Evaluating functions given their formula | Review for Ready; practices substituting inputs rather than deriving arithmetic-sequence rules. |
| Khan Academy — Intro to arithmetic sequence formulas | Explains recursive and explicit arithmetic rules and how the initial term affects the formula. |
Teacher/parent review of the source packet
No substantial mathematical issues were found in this section.
Prompts and data adapted from Mathematics Vision Project, Secondary Mathematics I, Module 1 student packet, Section 1.4, pages 15–18, licensed under CC BY 4.0. Explanations and diagrams prepared for this supplement.
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