Identifying Geometric Sequences by Checking Consecutive Ratios
Grade 9 students in California Reveal Math Algebra 1 learn to identify geometric sequences by computing every consecutive ratio and checking that all ratios are equal. A sequence is geometric if and only if a2/a1 = a3/a2 = a4/a3 = r for the same nonzero constant r. For example, 3,6,12,24 is geometric with r=2 (each ratio equals 2); 2,6,10,14 is not geometric (ratios are 3, 5/3, 7/5 — not equal); and 81,-27,9,-3 is geometric with r=-1/3. If even one ratio differs, the sequence fails the test.
Key Concepts
A sequence $a 1, a 2, a 3, \ldots$ is geometric if and only if every consecutive ratio is equal to the same nonzero constant $r$:.
$$\frac{a 2}{a 1} = \frac{a 3}{a 2} = \frac{a 4}{a 3} = \cdots = r$$.
Common Questions
How do you check if a sequence is geometric?
Divide each term by the term immediately before it and compare all resulting ratios. If every consecutive ratio equals the same nonzero constant r, the sequence is geometric.
Is 3, 6, 12, 24 a geometric sequence?
Yes. Computing 6/3=2, 12/6=2, 24/12=2. All ratios equal 2, so the sequence is geometric with common ratio r=2.
Is 2, 6, 10, 14 a geometric sequence?
No. Computing 6/2=3, 10/6=5/3, 14/10=7/5. The ratios are not equal, so the sequence is not geometric. It is actually arithmetic with a common difference of 4.
Is 81, -27, 9, -3 a geometric sequence?
Yes. Each ratio is -27/81=-1/3, 9/-27=-1/3, -3/9=-1/3. All equal -1/3, so the sequence is geometric with common ratio r=-1/3.
What is the common mistake when checking sequences?
Confusing geometric sequences (equal ratios) with arithmetic sequences (equal differences). Always compute ratios, not differences, to check if a sequence is geometric.
Which unit covers geometric sequences in Algebra 1?
This skill is from Unit 8: Exponential Functions in California Reveal Math Algebra 1, Grade 9.