Geometric Sequences
I can find the common ratio of a geometric sequence and write a rule for any term.
The problem this sheet solves
Students reach for a difference because arithmetic sequences came first, and a single subtraction does not reveal that the pattern is multiplicative. Testing only one pair is the root cause: a difference or a ratio computed once always produces a constant, and constants feel like confirmation. This sheet requires the ratio to be checked on a SECOND pair before it is used, which turns identifying the sequence type into a test rather than a guess. Arithmetic and geometric sequences are mixed so students must decide which they have, and the sheet includes patterns that are neither.
The key rule
Divide to find r, then an = a1 · rn − 1. Check r on a SECOND pair before you use it: a constant difference means arithmetic, a constant ratio means geometric, and two terms alone mean neither.
Divide, do not subtract. Then check a second gap.
Watch out
The mistake: For 3, 12, 48, 192, ... a student computes 12 − 3 = 9 and predicts 3, 12, 21, 30.
The fix: Divide: 12 ÷ 3 = 4 and 48 ÷ 12 = 4, so r = 4. Their own second gap disproves them — 48 − 12 is 36, not 9. A difference that changes is not a rule.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.