ACT Math: Sequences
Sequences is a Functions-domain topic on the ACT, testing your ability to spot a pattern and express it as a formula. This free, complete lesson covers arithmetic sequences, geometric sequences, recursive formulas, and the sum of finite arithmetic and geometric series. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Arithmetic Sequences
An arithmetic sequence has a constant difference, called the common difference (d), between consecutive terms. Its explicit (nth-term) formula is:
an = a1 + (n − 1)d
Worked Example: Find the 15th term of the arithmetic sequence 4, 9, 14, 19, ...
The common difference is d = 5.
a15 = 4 + (15 − 1)(5) = 4 + 70 = 74
1. Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...
Explanation: d = 4. a10 = 3 + (10 − 1)(4) = 3 + 36 = 39.
2. Find the common difference of the sequence 20, 14, 8, 2, ...
Explanation: d = 14 − 20 = −6 (each term decreases by 6).
3. Write an explicit formula for the sequence 5, 8, 11, 14, ...
Explanation: a1 = 5, d = 3. an = 5 + (n − 1)(3) = 5 + 3n − 3 = 3n + 2.
4. In an arithmetic sequence, the 3rd term is 17 and the 7th term is 37. What is the common difference?
Explanation: Going from the 3rd term to the 7th term takes 4 steps and changes the value by 37 − 17 = 20. d = 20 ÷ 4 = 5.
5. Using the sequence from the previous problem (3rd term = 17, common difference = 5), what is the 1st term?
Explanation: a3 = a1 + 2d, so 17 = a1 + 2(5), giving a1 = 17 − 10 = 7.
2. Geometric Sequences
A geometric sequence has a constant ratio, called the common ratio (r), between consecutive terms. Its explicit formula is:
an = a1 × rn − 1
Worked Example: Find the 6th term of the geometric sequence 3, 6, 12, 24, ...
The common ratio is r = 2.
a6 = 3 × 2⁵ = 3 × 32 = 96
1. Find the common ratio of the sequence 5, 15, 45, 135, ...
Explanation: r = 15 ÷ 5 = 3.
2. Find the 5th term of the geometric sequence 2, 6, 18, 54, ...
Explanation: r = 3. a5 = 2 × 3⁴ = 2 × 81 = 162.
3. Write an explicit formula for the sequence 4, 8, 16, 32, ...
Explanation: a1 = 4 and r = 2, giving an = 4 × 2n − 1.
4. A geometric sequence has first term 5 and common ratio 12. What is the 4th term?
Explanation: a4 = 5 × 123 = 5 × 18 = 58.
5. In a geometric sequence, the 2nd term is 6 and the 3rd term is 18. What is the common ratio?
Explanation: r = 18 ÷ 6 = 3.
3. Recursive Formulas
A recursive formula defines each term using the previous term, rather than a direct formula in terms of n. Arithmetic sequences follow an = an−1 + d; geometric sequences follow an = r × an−1.
Worked Example: A sequence is defined recursively by a1 = 7, an = an−1 + 4 for n ≥ 2. Find the first 4 terms.
a1 = 7, a2 = 7 + 4 = 11, a3 = 11 + 4 = 15, a4 = 15 + 4 = 19
1. A sequence is defined by a1 = 10, an = an−1 − 3. What is a4?
Explanation: a1 = 10, a2 = 7, a3 = 4, a4 = 1.
2. A sequence is defined by a1 = 2, an = 3 × an−1. What is a4?
Explanation: a1 = 2, a2 = 6, a3 = 18, a4 = 54.
3. The recursive formula an = an−1 + d describes what type of sequence?
Explanation: Adding a constant value (d) to get the next term is the defining feature of an arithmetic sequence.
4. The recursive formula an = r × an−1 describes what type of sequence?
Explanation: Multiplying by a constant ratio (r) to get the next term is the defining feature of a geometric sequence.
4. Sum of a Finite Series
The sum of the first n terms of an arithmetic sequence:
Sn = n2(a1 + an)
For a short geometric sequence, it's often just as fast to add the terms directly.
Worked Example: Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...
a10 = 3 + (10 − 1)(4) = 39.
S10 = 102(3 + 39) = 5(42) = 210
1. Find the sum of the first 20 terms of the arithmetic sequence 5, 9, 13, 17, ...
Explanation: a20 = 5 + 19(4) = 81. S20 = 202(5 + 81) = 10(86) = 860.
2. Find the sum of the first 15 terms of an arithmetic sequence with a1 = 2 and d = 3.
Explanation: a15 = 2 + 14(3) = 44. S15 = 152(2 + 44) = 7.5(46) = 345.
3. What is the sum of the integers from 1 to 100?
Explanation: This is an arithmetic sequence with a1 = 1, a100 = 100, and n = 100: S = 1002(1 + 100) = 50(101) = 5,050.
4. Find the sum of the first 4 terms of the geometric sequence 3, 6, 12, 24.
Explanation: 3 + 6 + 12 + 24 = 45.
5. Find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, 162.
Explanation: 2 + 6 + 18 + 54 + 162 = 242.
Common Mistakes to Avoid
- Using n instead of (n − 1) in the explicit formula. Both the arithmetic and geometric formulas use (n − 1), since the first term itself doesn't require any steps of change applied to it yet.
- Confusing an arithmetic and a geometric sequence. Check whether consecutive terms share a constant difference (arithmetic) or a constant ratio (geometric), the two patterns look very different once you check the actual numbers.
- Assuming every sequence with growing terms is geometric. An arithmetic sequence with a large common difference can also grow quickly; the defining test is always constant difference versus constant ratio, not how fast the numbers increase.
- Mixing up a recursive formula with an explicit formula. A recursive formula needs the previous term to compute the next one; an explicit formula computes any term directly from n, without needing prior terms.
- Forgetting that the arithmetic sum formula needs the last term of that specific sum, not the sequence's general nth term formula. Find an for the exact number of terms you're summing before plugging into the sum formula.
Frequently Asked Questions
How can I quickly tell if a sequence is arithmetic, geometric, or neither?
Subtract consecutive terms; if the difference is always the same, it's arithmetic. If that fails, divide consecutive terms; if the ratio is always the same, it's geometric. If neither pattern holds, the sequence follows some other rule.
Do I need to memorize the sum formulas, or can I just add the terms directly?
For a small number of terms, adding directly is often just as fast, especially for geometric sequences. For larger arithmetic sequences (dozens or hundreds of terms), the formula Sn = n2(a1 + an) is far quicker and worth knowing.
What real-world situations use geometric sequences?
Anything involving repeated percentage growth or decay follows a geometric pattern: compound interest, population growth at a constant rate, or a substance's radioactive decay, since each step multiplies the previous amount by the same factor.
Where can I practice more problems like these?
The Sequences quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.