ACT Math Sequences and Series: Identify Patterns and Solve in Seconds

Type 1: Arithmetic sequence. The difference between consecutive terms is constant (called the common difference d). Example: 2, 5, 8, 11 (common difference is 3). Formula: a_n=a_1+(n-1)d, where a_n is the nth term. Type 2: Geometric sequence. The ratio between consecutive terms is constant (called the common ratio r). Example: 2, 6, 18, 54 (common ratio is 3). Formula: a_n=a_1×r^(n-1). Identify which type you have, apply the formula, and you've solved the problem. ACT rarely asks for anything beyond finding a term or the sum.

Quick identification: In 2, 5, 8, 11, each term increases by 3 (arithmetic, d=3). In 2, 6, 18, 54, each term is multiplied by 3 (geometric, r=3). Once you spot which type, the formula does the work.

Mistake 1: Confusing arithmetic and geometric sequences. If terms increase by the same amount, it's arithmetic. If terms are multiplied by the same factor, it's geometric. Mistake 2: Miscalculating the common difference or ratio. For arithmetic, verify d by subtracting two consecutive terms. For geometric, verify r by dividing two consecutive terms. Mistake 3: Using the wrong formula or plugging in wrong values. For a_n=a_1+(n-1)d, remember a_1 is the first term and n is the term number you're finding. Double-check that you've identified a_1 and d (or r) correctly before applying the formula.

Create a card showing both formulas and one worked example of each type. Reference it daily this week. By test day, both formulas will be automatic.

Arithmetic Problem 1: Find the 10th term of 3, 7, 11, 15... Common difference d=4. a_10=3+(10-1)(4)=3+36=39. Arithmetic Problem 2: Find the 5th term of 100, 90, 80, 70... Common difference d=-10. a_5=100+(5-1)(-10)=100-40=60. Geometric Problem 1: Find the 4th term of 2, 6, 18, 54... Common ratio r=3. a_4=2×3^(4-1)=2×27=54. Geometric Problem 2: Find the 5th term of 1000, 100, 10, 1... Common ratio r=1/10 (or 0.1). a_5=1000×(0.1)^(5-1)=1000×0.0001=0.1. Problem 3 (Mixed): Is 1, 3, 9, 27... arithmetic or geometric? Geometric (r=3). Problem 4 (Mixed): Is 5, 10, 15, 20... arithmetic or geometric? Arithmetic (d=5). Identify the type and formula for each, then solve.

Find six sequence questions from a practice test. For each, identify the type, extract the common difference or ratio, and apply the formula. By the sixth question, sequence pattern identification will feel automatic.

Sequence questions appear on most ACT Math tests, usually in questions 35-50. Once you identify the sequence type, the formula does most of the work. Students who master the two sequence types pick up 1-2 points on the math section because pattern identification is reliable once you know the two categories and their formulas.

Drill sequence identification daily this week. Each day, identify whether five sequences are arithmetic or geometric, then apply the formula. By test day, you should recognize any sequence type and solve in under 60 seconds.