ACT Math: Identify and Solve Arithmetic and Geometric Sequences Quickly

Arithmetic sequence: adds a constant difference (d) each term. Example: 2, 5, 8, 11 (adds 3 each time). Formula: a_n=a_1+(n-1)d. Geometric sequence: multiplies by a constant ratio (r) each term. Example: 2, 6, 18, 54 (multiplies by 3 each time). Formula: a_n=a_1*r^(n-1). To identify which type, calculate the difference (arithmetic) or ratio (geometric) between consecutive terms. If the difference is constant, use arithmetic. If the ratio is constant, use geometric. These two formulas unlock every sequence problem on ACT Math; knowing them cold gives you a huge advantage.

Example: Find the 10th term of 3, 7, 11, 15... Difference: 4 (arithmetic). d=4, a_1=3. a_10=3+(10-1)*4=3+36=39. Example: Find the 5th term of 2, 6, 18... Ratio: 3 (geometric). r=3, a_1=2. a_5=2*3^(5-1)=2*81=162. Each formula gives the answer in 30 seconds.

Trap 1: Confusing arithmetic and geometric. Test both: calculate differences and ratios. If differences are constant, it's arithmetic. If ratios are constant, it's geometric. Trap 2: Forgetting to use the formula correctly. The formula a_n=a_1+(n-1)d uses (n-1), not n. The formula a_n=a_1*r^(n-1) uses exponent (n-1), not r^n. Writing n instead of (n-1) gives wrong answers. Double-check your formula before substituting values; make sure you're using (n-1) as the exponent or multiplier, not n.

Before you calculate, verify: Is this arithmetic or geometric? Write down the formula you'll use. Substitute values. Check your answer by verifying it fits the pattern (add d or multiply by r from the previous term).

Problem 1: Find the 7th term of 5, 10, 15, 20... Type: Arithmetic (difference=5). a_7=5+(7-1)*5=5+30=35. Problem 2: Find the 6th term of 1, 4, 16, 64... Type: Geometric (ratio=4). a_6=1*4^(6-1)=4^5=1024. Problem 3: An arithmetic sequence has a_1=10 and d=3. Find a_20. a_20=10+(20-1)*3=10+57=67. Problem 4: A geometric sequence has a_1=2 and r=2. Find a_8. a_8=2*2^(8-1)=2*128=256. All four problems use the same formulas; only the numbers and sequence type change.

Do ten more sequence problems daily for one week. By test day, you'll identify sequence types and apply formulas so fast that sequence problems feel like free points.

Sequence problems appear on most ACT Math sections and are typically easy-to-medium difficulty. These are points you should get because the formulas are straightforward. Students who know the arithmetic and geometric sequence formulas solve these problems in 30 seconds each; students who don't spend 3-5 minutes trying to figure out the pattern.

Learn these two formulas this week. Drill them daily. By test day, sequence formulas will be so automatic that you'll use them without thinking, freeing mental energy for harder problems.