Sequences appear regularly on the ACT but rarely on the SAT. Here is everything you need to know.
Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms.
Example: has , .
The 50th term: .
Sum of first terms:
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms.
Example: has , .
The 10th term: .
Sum of first terms:
How to Identify the Type
Given a sequence, test both:
- Subtract consecutive terms. Constant difference? Arithmetic.
- Divide consecutive terms. Constant ratio? Geometric.
If neither is constant, it is neither arithmetic nor geometric.
Finding Missing Terms
If the ACT gives you two terms and their positions:
Arithmetic: and are given.
Then find .
Common Mistakes to Avoid
- Off-by-one errors: The formula uses , not . The 1st term uses .
- Confusing and : Arithmetic = add . Geometric = multiply by .
- Geometric with negative ratio: has . The signs alternate.
ACT Pro Tip
When the ACT gives you the first few terms of a sequence and asks for a later term, first determine if it is arithmetic or geometric. Then plug into the appropriate formula. Do not try to extend the sequence by hand — for the 50th term, the formula is the only practical approach.
Practice sequences with our Sequences lessons — ACT-exclusive content.