How to Solve Number Sequences: 4 Steps for Spotting the Rule

"2, 5, 11, 23, … what comes next?" Number sequence problems are a staple of IQ tests and aptitude assessments. Rather than staring at the numbers, deciding in advance what to check — and in what order — makes a big difference to both accuracy and speed. This article organizes it into four steps.

Step 1: Take the differences (arithmetic and second differences)

First, write out the differences between neighboring terms. If the difference is constant, it is an arithmetic sequence (e.g., 3, 7, 11, 15 → +4). If the differences themselves change, check whether the sequence of differences follows its own rule (second differences). For example, 2, 3, 5, 8, 12 has differences of +1, +2, +3, +4. In practice, taking differences alone resolves a large share of sequence problems.

Step 2: Take the ratios (geometric and multiplicative patterns)

If the differences show no rule, divide each term by the one before it. If the ratio is constant, as in 2, 6, 18, 54 (×3), it is a geometric sequence. Compound patterns like "×2+1" or "×2−1" also surface here. The opening sequence 2, 5, 11, 23 has differences of 3, 6, 12 — doubling each time — and can equally be solved as "×2+1" (2→5→11→23→47). When multiple readings converge on the same answer, you can be confident.

Step 3: Read every other term (alternating patterns)

If neither differences nor ratios reveal a rule, split the sequence into odd- and even-positioned terms. 1, 10, 3, 20, 5, 30 is two interleaved sequences: "1, 3, 5" and "10, 20, 30." This type — two independent rules woven together — is baffling if you don't know it, and takes seconds if you do.

Step 4: Suspect a link to the position number

As a last resort, check whether each term is determined by its position. 1, 4, 9, 16 is the square of the position number; 2, 6, 12, 20 is n×(n+1). Matrix reasoning uses the same idea (row number × column number) — the thinking is shared with how to solve matrix reasoning problems.

Practical tips for speed

Under time pressure, check mechanically in the order differences → ratios → alternating → position. And always verify the rule you find against every term through the last one; if even one term doesn't fit, it is a different rule. Pattern recognition improves with practice on similar problems. Try the worked examples on the pattern recognition problem page, then measure your overall tendencies, including numerical reasoning, with the free IQ test.