Sequences Arithmetic and Geometric — ACT Math Guide
Sequences arithmetic and geometric ACT questions test your ability to identify patterns and use formulas to find specific terms or sums. These problems involve recognizing whether a sequence increases by a constant difference (arithmetic) or constant ratio (geometric). The ACT math section typically includes 2-3 sequence questions out of 60 total questions, making them worth mastering for your ACT math score. With the right approach, these questions become some of the most predictable points you can earn.
What You Need to Know
Arithmetic sequence: Each term increases by the same amount (common difference)
Geometric sequence: Each term is multiplied by the same number (common ratio)
nth term formula: Different formulas for arithmetic vs geometric sequences
Sum formulas: Calculate the total of multiple terms in a sequence
Pattern recognition: Identify sequence type from given terms
📐 KEY FORMULAS:
Arithmetic: aₙ = a₁ + (n-1)d
Geometric: aₙ = a₁ × r^(n-1)
⏱️ ACT TIME TIP: Check if differences are equal (arithmetic) or ratios are equal (geometric) — this takes 10 seconds and saves you from using wrong formulas.
How to Solve Sequences on the ACT Math Section
Example Question 1 — Easy/Medium Difficulty
In the arithmetic sequence 7, 12, 17, 22, ..., what is the 15th term?
A) 77
B) 82
C) 87
D) 92
E) 97
Solution:
Step 1: Identify this as arithmetic (constant difference of +5)
Step 2: Use formula aₙ = a₁ + (n-1)d where a₁ = 7, d = 5, n = 15
Step 3: a₁₅ = 7 + (15-1)(5) = 7 + 14(5) = 7 + 70 = 77
✅Answer: A — The 15th term equals 77 using the arithmetic sequence formula.
Example Question 2 — Hard Difficulty
The first term of a geometric sequence is 3, and the fourth term is 24. What is the seventh term?
A) 96
B) 144
C) 192
D) 216
E) 384
Solution:
Step 1: Use aₙ = a₁ × r^(n-1) to find the common ratio
Step 2: 24 = 3 × r^(4-1) → 24 = 3r³ → r³ = 8 → r = 2
Step 3: Find a₇ = 3 × 2^(7-1) = 3 × 2⁶ = 3 × 64 = 192
✅Answer: C — With common ratio 2, the seventh term equals 192.
Common ACT Math Mistakes to Avoid
❌Mistake: Confusing arithmetic and geometric sequence formulas
✅Fix: Always check differences first (arithmetic) then ratios (geometric)
❌Mistake: Using the wrong value for n in the formula
✅Fix: Remember n represents the position number, not the number of steps
❌Mistake: Forgetting to subtract 1 in the exponent for geometric sequences
✅Fix: The formula is r^(n-1), not r^n
❌Mistake: Making arithmetic errors with negative common differences or ratios
✅Fix: Use your calculator for all calculations — it's allowed throughout ACT math
Practice Question — Try It Yourself
In the sequence 2, 6, 18, 54, ..., what is the sum of the first 5 terms?
A) 162
B) 242
C) 324
D) 486
E) 728
Show Answer
Answer: B — This is geometric with r = 3. The 5th term is 2 × 3⁴ = 162. Sum = 2 + 6 + 18 + 54 + 162 = 242.
Key Takeaways for the ACT
Spend 15 seconds identifying sequence type before applying formulas
ACT math sequences questions often give you the first few terms — use them to find patterns
Your calculator handles all the arithmetic, so focus on setting up the right formula
Geometric sequences with fractional ratios appear less frequently but follow the same rules
Most ACT sequence questions ask for a specific term, not a sum
Related ACT Math Topics
Strengthen your ACT math prep with these related topics:
Quadratic equations →
Exponential functions →