Introduction
Sequence questions are a common feature on the ACT® Math Test, often appearing multiple times per test across all difficulty levels. Understanding these concepts is crucial for success on the exam.
Let's look at the basics of arithmetic and geometric sequences to help you answer these questions with confidence.
What are Arithmetic and Geometric Sequences?
Arithmetic Sequence: Adding
Geometric Sequence: Multiplying
These simple definitions are key to understanding how sequences work on the ACT® Math test.
How Sequences Appear on the ACT® Math Test
The ACT® Math section typically presents sequence questions in two ways:
- They might explicitly state that a sequence is arithmetic or geometric.
- They might provide a series of numbers and ask you to determine the pattern.
If the sequence type isn't specified, your job is to figure out whether the numbers increase by addition or multiplication. Then, you'll need to:
- For arithmetic sequences: Subtract consecutive terms to find the common difference.
- For geometric sequences: Divide consecutive terms to find the common ratio.
Once you've identified the pattern, you can apply this logic to find the next terms or solve related problems.
Practice Questions
Question:
Example 1: What is the next term in the arithmetic sequence 3, 7, 11, 15, ...?
- A) 17
- B) 18
- C) 19
- D) 20
- E) 21
Click for the Answer
Correct Answer: C) 19
Explanation:
The common difference is 4 (7 - 3 = 4, 11 - 7 = 4, etc.). The next term is 15 + 4 = 19.
Question:
Example 2: What is the next term in the geometric sequence 2, 6, 18, 54, ...?
- A) 108
- B) 162
- C) 216
- D) 324
- E) 432
Click for the Answer
Correct Answer: B) 162
Explanation:
The common ratio is 3 (6 / 2 = 3, 18 / 6 = 3, etc.). The next term is 54 * 3 = 162.
Question:
Example 3: What is the 5th term in the arithmetic sequence where the first term is 5 and the common difference is 7?
- A) 26
- B) 29
- C) 32
- D) 33
- E) 35
Click for the Answer
Correct Answer: D) 33
Explanation:
The sequence is 5, 12, 19, 26, 33 (common difference is 7). The 5th term is 5 + 4 * 7 = 33.
Conclusion
Sequences are all about recognizing patterns and applying them consistently, whether addition or multiplication.