K-12 Math (US): 9-12 Algebra K-12 Math Sequences Arithmetic vs geometric

Study Guide: Sequences — Arithmetic vs. Geometric

Grade 9–12 | Algebra

1. The Driving Question

If you save $10 every week, your savings grow in a predictable way—but if you instead double your savings each week, the numbers explode. Why does one pattern add up steadily while the other turns into a rocket? And how can you tell, just by looking at a sequence, whether it’s built on addition or multiplication… or something else entirely?

2. The Core Idea — Built, Not Listed

Imagine two video game characters leveling up. Alex gains 5 XP every time they complete a quest—no matter how strong they already are. Their XP looks like this: 10, 15, 20, 25, 30… Each step is the same size, like climbing a ladder with evenly spaced rungs. This is an arithmetic sequence, where the difference between terms (the common difference) stays constant.

Now meet Blaze, whose XP doubles after every quest. Their XP starts at 3 and grows like this: 3, 6, 12, 24, 48… The gaps between numbers get wider and wider, like a snowball rolling downhill. This is a geometric sequence, where each term is multiplied by the same number (the common ratio).

The key difference? Arithmetic sequences are about adding the same amount, while geometric sequences are about multiplying by the same factor. Both are predictable, but their growth feels totally different—one is steady, the other exponential.

Key Vocabulary:
- Sequence: An ordered list of numbers where each term follows a rule.
Example: The sequence of years the Summer Olympics are held (1996, 2000, 2004, …) has a common difference of 4.
Note for college: In higher math, sequences can be defined recursively (each term depends on the previous one) or explicitly (a formula for the nth term).

• Common difference (d): The constant amount added to get the next term in an arithmetic sequence.
  Example: In the sequence of U.S. presidential election years (2000, 2004, 2008, …), d = 4.
  Note for college: The common difference can be negative (e.g., -3 in 10, 7, 4, 1, …), representing decay.

• Common ratio (r): The constant factor multiplied to get the next term in a geometric sequence.
  Example: If a rumor spreads so that each person tells 3 new people, the number of people who know the rumor each hour forms a geometric sequence with r = 3 (1, 3, 9, 27, …).
  Note for college: The common ratio can be a fraction (e.g., r = 1/2 in 16, 8, 4, 2, …), representing halving or decay.

• Explicit formula: A rule that gives the nth term directly, without needing the previous term.
  Example: The arithmetic sequence 5, 9, 13, 17, … has the explicit formula aₙ = 5 + 4(n – 1).
  Note for college: Explicit formulas are critical in calculus for defining sequences and series (e.g., Taylor series).

3. Assessment Translation

How this appears on assessments:
- Multiple choice: Identify whether a sequence is arithmetic, geometric, or neither, and find the common difference/ratio.
Distractor patterns: - Confusing addition with multiplication (e.g., seeing 2, 4, 8, 16 and assuming d = 2 instead of r = 2).
- Misidentifying the first term (e.g., thinking the sequence 3, 6, 12, … starts at 6).
- Overlooking negative common differences/ratios (e.g., -5, -10, -20, … has r = 2, not -2).

• Short answer/constructed response: Write the explicit formula for a sequence or find a specific term.
  What a "proficient" response looks like:
• Prompt: Find the 10th term of the geometric sequence 3, 6, 12, 24, …

• Proficient response:
  > This is a geometric sequence with first term a₁ = 3 and common ratio r = 6/3 = 2. The explicit formula is aₙ = 3(2)ⁿ⁻¹. For the 10th term, plug in n = 10:
  > a₁₀ = 3(2)⁹ = 3(512) = 1,536.

• SAT/ACT framing:

• SAT Math: Often tests identifying sequences in word problems (e.g., "A bacteria culture doubles every hour. If there are 100 bacteria at noon, how many are there at 3 PM?").
• ACT Math: May ask for the nth term or the sum of the first n terms (e.g., "What is the sum of the first 5 terms of the arithmetic sequence 4, 7, 10, …?").

Model student response (proficient level):
Prompt: Determine whether the sequence 1, 4, 9, 16, 25, … is arithmetic, geometric, or neither. Justify your answer.
Response:

This sequence is neither arithmetic nor geometric. For it to be arithmetic, the difference between terms would have to be constant. Here, the differences are 3, 5, 7, 9, …, which change. For it to be geometric, the ratio between terms would have to be constant. The ratios are 4/1 = 4, 9/4 = 2.25, 16/9 ≈ 1.78, …, which also change. This sequence is actually the squares of the counting numbers (n²).

4. Mistake Taxonomy

Mistake 1: Misidentifying the sequence type
- Prompt: Is the sequence 2, 4, 8, 16, … arithmetic or geometric? - Common wrong response: "Arithmetic, because you add 2, then 4, then 8." - Why it loses credit: The student is looking at the differences between differences (2, 4, 8) instead of the differences themselves (2, 4, 8) or the ratios (2, 2, 2). Arithmetic sequences require a constant difference, not a changing one.
- Correct approach: - Calculate the differences: 4 – 2 = 2, 8 – 4 = 4, 16 – 8 = 8 → Not constant.
- Calculate the ratios: 4/2 = 2, 8/4 = 2, 16/8 = 2 → Constant ratio of 2.
- Conclusion: Geometric sequence with r = 2.

Mistake 2: Incorrect explicit formula
- Prompt: Write the explicit formula for the arithmetic sequence 10, 7, 4, 1, … - Common wrong response: aₙ = 10 + 3(n – 1) - Why it loses credit: The student used the wrong sign for the common difference. The sequence decreases by 3 each time, so d = -3, not +3.
- Correct approach: - First term a₁ = 10.
- Common difference d = 7 – 10 = -3.
- Explicit formula: aₙ = 10 + (-3)(n – 1) = 10 – 3(n – 1).
- Simplified: aₙ = 13 – 3n.

Mistake 3: Confusing recursive and explicit formulas
- Prompt: A geometric sequence has a first term of 5 and a common ratio of 3. Write a recursive formula for the sequence.
- Common wrong response: aₙ = 5(3)ⁿ⁻¹ (this is the explicit formula).
- Why it loses credit: The student provided the explicit formula instead of the recursive one. Recursive formulas define each term based on the previous term.
- Correct approach: - Recursive formula: a₁ = 5, aₙ = 3 · aₙ₋₁ for n > 1.
- Explicit formula (for reference): aₙ = 5(3)ⁿ⁻¹.

5. Connection Layer

1. Within math: Sequences → Series — A series is the sum of a sequence’s terms. Understanding whether a sequence is arithmetic or geometric tells you which formula to use for its sum (e.g., Sₙ = n/2(a₁ + aₙ) for arithmetic, Sₙ = a₁(1 – rⁿ)/(1 – r) for geometric).

2. Across subjects: Sequences → Biology (population growth) — Geometric sequences model exponential growth (e.g., bacteria doubling every hour), while arithmetic sequences model linear growth (e.g., a tree adding the same number of rings each year).

3. Outside school: Sequences → Music (tuning systems) — The notes on a piano follow a geometric sequence: each key’s frequency is r = 2^(1/12) times the previous one (the "12-tone equal temperament" system). This is why octaves sound "the same" but higher or lower.

6. The Stretch Question

If you add the same number to every term in an arithmetic sequence, is the result still arithmetic? What if you multiply every term by the same number? What if you do both?

Pointer toward the answer: - Adding a constant to every term of an arithmetic sequence preserves the common difference (e.g., 2, 5, 8, … + 3 = 5, 8, 11, … still has d = 3). The sequence remains arithmetic.
- Multiplying every term by a constant also preserves the common difference (e.g., 2, 5, 8, … × 2 = 4, 10, 16, … has d = 6, which is 2 × 3). The sequence remains arithmetic.
- Doing both (adding k and multiplying by m) still leaves the sequence arithmetic, but the common difference becomes m × d.
- For geometric sequences, multiplying by a constant preserves the common ratio, but adding a constant breaks the geometric property (e.g., 3, 6, 12, … + 1 = 4, 7, 13, … is no longer geometric). This is why geometric sequences are more "fragile" under addition.