Unit 3 · Topic 2.1
Change in Arithmetic and Geometric Sequences
Compare additive change in arithmetic sequences with multiplicative change in geometric sequences. Develop the idea through symbolic, numerical, graphical, and contextual representations.
Learning Goals
- Treat a sequence as a function on whole-number indices.
- Write recursive and explicit arithmetic and geometric rules.
- Distinguish constant differences from constant ratios in multiple representations.
1. Essential Structure
Compare additive change in arithmetic sequences with multiplicative change in geometric sequences.
Read the formula together with its domain, units, starting input, and the interval length over which change is measured.
2. Core Ideas
- Arithmetic change is additive: \(a_{n+1}-a_n=d\).
- Geometric change is multiplicative: \(g_{n+1}/g_n=r\), when the denominator is nonzero.
- Known-term forms are \(a_n=a_k+d(n-k)\) and \(g_n=g_kr^{n-k}\).
- Sequence graphs are discrete because their domains are index values, not all real numbers.
3. Graph and Representation
Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.
4. Original Worked Example
An arithmetic sequence has \(a_1=5\) and \(d=3\). Find \(a_4\).
Check the result in the original representation and explain its meaning, including units when the quantities are contextual.
5. AP Reasoning Workflow
- Identify the starting index and domain.
- Calculate consecutive differences and ratios.
- Choose the pattern supported by a constant value.
- Anchor the rule at a known term.
- Check another term and interpret the parameters.
A strong AP response shows the mathematical evidence first and then states a precise conclusion.
6. Extended Example and Application
A balance is 240 dollars at month 3 and then adds 35 dollars monthly. Its arithmetic model is \(A_n=240+35(n-3)\). If it instead grows 4% monthly, the geometric model is \(G_n=240(1.04)^{n-3}\).
7. Technology and Validation
A scatterplot can reveal collinear or exponential-looking discrete points, but calculated differences or ratios provide the numerical justification.
When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.
8. A Sequence Is a Function
A sequence assigns one real output to each allowed index. Its domain is usually \(\{0,1,2,\ldots\}\), \(\{1,2,3,\ldots\}\), or a finite subset of those values. That is why its graph contains separate points.
| Index \(n\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(A_n=7+4n\) | 7 | 11 | 15 | 19 | 23 |
| \(G_n=7(1.5)^n\) | 7 | 10.5 | 15.75 | 23.625 | 35.4375 |
The arithmetic outputs have constant first difference 4. The geometric outputs have constant ratio 1.5. Connecting the points may suggest a related real-valued function, but the connecting inputs are not members of the sequence domain.
9. Recursive and Explicit Rules
A recursive rule explains how to move from one term to the next, so it must include an initial term. An explicit rule calculates a distant term directly from its index.
| Pattern | Recursive form | Explicit form, starting at \(n=1\) |
|---|---|---|
| Arithmetic | \(a_1=A,\ a_n=a_{n-1}+d\) | \(a_n=A+(n-1)d\) |
| Geometric | \(g_1=G,\ g_n=rg_{n-1}\) | \(g_n=Gr^{n-1}\) |
The exponent \(n-1\) counts how many multiplications carry the first term to the nth term. If indexing begins at zero, the corresponding forms become \(a_n=a_0+nd\) and \(g_n=g_0r^n\).
10. Identify the Pattern from Data
Use equal index steps and test both structures. For outputs 48, 36, 27, and 20.25, the differences are not constant, but every consecutive ratio equals \(3/4\). Therefore the data are geometric.
A negative common difference produces a decreasing arithmetic sequence. A geometric sequence with \(0<r<1\) and positive terms decays toward zero. A negative common ratio alternates signs, so “increasing” and “decreasing” may not describe the entire sequence.
11. Recover a Rule from Two Terms
Arithmetic example: Suppose \(a_3=14\) and \(a_8=34\). Five index steps produce an output change of 20, so \(d=20/5=4\). Moving back two steps gives \(a_1=14-2(4)=6\).
Geometric example: Suppose \(g_2=6\) and \(g_5=162\). Three index steps multiply the output by 27, so \(r^3=27\) and \(r=3\). Then \(g_1=6/3=2\).
If the index gap is even, an equation such as \(r^2=4\) can support both \(r=2\) and \(r=-2\). Use additional terms or contextual restrictions before selecting a ratio.
12. Compare Absolute and Proportional Change
An increasing arithmetic sequence adds the same absolute amount each step. An increasing positive geometric sequence adds larger absolute amounts because the same ratio is applied to a growing term.
| Question | Arithmetic evidence | Geometric evidence |
|---|---|---|
| What stays constant? | \(a_{n+1}-a_n=d\) | \(g_{n+1}/g_n=r\) |
| What is the graph shape? | Discrete points on a line | Discrete points on an exponential pattern |
| How do first differences behave? | They stay equal | They scale with the current term |
| Related continuous family | Linear | Exponential when \(r>0\) |
13. Context, Units, and Domain
A theater has 18 seats in its first row and 3 more seats in each successive row. If the theater has 20 rows, then \(S_n=18+3(n-1)\) has domain \(1\le n\le20\), where \(n\) is a whole number. The model should not be evaluated at row 4.6 or beyond row 20.
A laboratory sample retains 82% of its amount after each completed hour. If the initial amount is 500 mg, then \(M_n=500(0.82)^n\) for whole-number hours. The ratio 0.82 means 82% remains and 18% is lost each step.
State what the index counts, the units of each term, the starting index, and whether the sequence is finite. These details are part of the model rather than optional decoration.
14. AP-Ready Justification
When classifying a sequence, name the numerical evidence and connect it to the model.
Arithmetic response: “Over equal one-index steps, each output difference is \(-5\), so the sequence has constant additive change and is arithmetic with common difference \(-5\).”
Geometric response: “Over equal one-index steps, each nonzero output is multiplied by 1.12, so the sequence has constant proportional change and is geometric with common ratio 1.12, representing 12% growth per step.”
Do not justify a choice only by saying that the points look linear or curved. Tables, formulas, and graphs should tell the same story.
15. Common Errors
- Do not confuse a common difference with a common ratio.
- Connecting sequence points as though every real input were allowed.
- Confusing a decimal growth rate \(r-1\) with the common ratio \(r\).
- Giving a numerical result without a domain check, units, or interpretation.
Key Takeaways
- Compare additive change in arithmetic sequences with multiplicative change in geometric sequences.
- Core relationship: \(a_n=a_1+(n-1)d,\qquad g_n=g_1r^{n-1}\)
- Error check: Do not confuse a common difference with a common ratio.
- An arithmetic sequence has \(a_2=18\) and \(a_6=42\). Find its common difference and explicit rule.
- A geometric sequence has \(g_2=12\) and \(g_5=96\). Find its common ratio.
- Explain why a sequence and a related continuous function can share a formula but have different domains.