ACT Math: Identify Patterns in Sequences in 30 Seconds Flat

Step 1: Check if the sequence is arithmetic (constant difference between consecutive terms). Subtract the second term from the first, then the third from the second. If both differences are the same, it's arithmetic. Example: 3, 7, 11, 15. Differences: 7-3=4, 11-7=4, 15-11=4. Arithmetic with common difference d=4. Step 2: If differences aren't constant, check if it's geometric (constant ratio between consecutive terms). Divide the second term by the first, then the third by the second. If both ratios match, it's geometric. Example: 2, 6, 18, 54. Ratios: 6/2=3, 18/6=3, 54/18=3. Geometric with common ratio r=3. This two-step method identifies 95% of sequences on the ACT in 20-30 seconds.

Why this matters: once you identify the pattern (arithmetic or geometric), you can find any term in the sequence using the formula. Arithmetic: nth term=first term+(n-1)×d. Geometric: nth term=first term×r^(n-1). The patterns unlock the formulas.

Trap 1: Assuming a sequence is arithmetic when it's actually something else. Example: 1, 4, 9, 16 (differences: 3, 5, 7; these don't match, so it's not arithmetic). Correct: recognize it's perfect squares. Trap 2: Forgetting to check the ratio for geometric sequences. Just because you see multiplication doesn't mean the ratio is constant. Example: 2, 4, 7 (ratios: 4/2=2, 7/4=1.75; ratios don't match, so it's neither arithmetic nor geometric). Trap 3: Mixing up arithmetic and geometric formulas. For arithmetic, use d (difference). For geometric, use r (ratio). Students lose points by applying the wrong formula to the wrong sequence type.

Cure: always apply the two-step test. First check differences; if they're constant, use arithmetic. If not, check ratios. This discipline prevents formula mix-ups and catches unusual sequences that don't fit either pattern.

Sequence 1: 5, 10, 20, 40. (Differences don't match; ratios all equal 2. Geometric with r=2. Next term: 80.) Sequence 2: 12, 9, 6, 3. (Differences all equal -3. Arithmetic with d=-3. Next term: 0.) Sequence 3: 1, 1, 2, 3, 5, 8. (Differences: 0,1,1,2,3—not constant. Ratios don't match. This is Fibonacci—not arithmetic or geometric. Recognize the pattern separately.) Sequence 4: 3, 6, 12, 24. (Ratios all equal 2. Geometric with r=2. Next term: 48.) Sequence 5: 100, 95, 90, 85. (Differences all equal -5. Arithmetic with d=-5. Next term: 80.) Four of the five are arithmetic or geometric; Sequence 3 is Fibonacci, which is neither, and you should recognize it as special.

Time yourself: you should classify and extend each sequence in under 30 seconds, including writing the formula. This speed comes from drilling the two-step test repeatedly.

Sequences appear in 1-2 ACT Math questions per section, sometimes hidden in word problems about population growth (geometric) or savings plans (arithmetic). If you recognize the pattern, you solve in 30 seconds using the formula. If you don't, you're stuck trying to extend the sequence by hand, wasting 3 minutes. Pattern recognition is a pure time-saving skill with zero additional difficulty; it's a gift to yourself once you master it.

Spend one week drilling the two-step test on 10-15 sequences from practice tests. By test day, you'll recognize patterns instantly and solve sequence problems faster than 95% of test-takers.