Run a fixed checklist on every series and you will rarely be stuck: look at the difference between terms, then the ratio, then squares and cubes, then whether two patterns alternate.
Series-type checklist
Constant or growing difference — 2, 6, 12, 20 (gaps 4, 6, 8). Constant ratio — 3, 6, 12, 24 (each ×2). Squares / cubes — 1, 4, 9, 16 or 1, 8, 27, 64. Alternating — two interleaved series sharing one list.
- Constant or growing difference — 2, 6, 12, 20 (gaps 4, 6, 8).
- Constant ratio — 3, 6, 12, 24 (each ×2).
- Squares / cubes — 1, 4, 9, 16 or 1, 8, 27, 64.
- Alternating — two interleaved series sharing one list.
How to Approach It
- Examine the differences — Look first at the gaps between consecutive terms. A constant gap means an arithmetic series; a growing gap may hide a second pattern inside the differences themselves.
- Test ratios, squares and cubes — If each term is a constant multiple of the last, it is geometric. If the terms resemble 1, 4, 9, 16 or 1, 8, 27, suspect squares or cubes at work.
- Check for an alternating pattern — Two interleaved series often share a single list. If no single rule fits, separate the odd and even positions and test each on its own.
- For letters, convert to positions — Translate A=1 through Z=26 and find the step on the numbers. The alphabet's gaps are easy to miscount by eye, so let the numbers do the work.
Techniques & Methods
- Difference ladder — Check first differences, then the differences of those. e.g. 2,6,12,20 → diffs 4,6,8 → next +10 → 30.
- Ratio test — A constant multiplier signals a geometric series. e.g. 3,6,12,24 → ×2.
- Squares / cubes radar — Memorise 1,4,9,16,25 and 1,8,27,64 to spot them instantly. e.g. 1,8,27,64 → next 125.
- Letter-to-number — Convert A=1 … Z=26, then read the step on the numbers. e.g. B,D,G,K → +2,+3,+4 → next +5 = P.
The Edge
For letter series, convert to position numbers (A=1 … Z=26). The hidden rule is almost always a simple step on those numbers — +2, -3, or alternating steps. If the differences are not constant, take the difference of the differences. A constant second difference signals a square-based pattern.Worked example
Find the next term: 2, 6, 12, 20, 30, __
- When no obvious multiple links the terms, look at the differences first: 6−2=4, 12−6=6, 20−12=8, 30−20=10.
- The differences themselves form a pattern: 4, 6, 8, 10 — increasing by 2 each time.
- So the next difference is 10 + 2 = 12.
- Add it to the last term: 30 + 12 = 42.
Answer: Next term = 42
Worked example
Find the next letter: A, C, E, G, __
- Letters are easiest handled as positions: A=1, C=3, E=5, G=7.
- Those positions go 1, 3, 5, 7 — a constant step of +2.
- The next position is 7 + 2 = 9.
- Position 9 is the letter I.
Answer: Next letter = I
Worked Drills
Worked example
Find the next term: 6, 11, 21, 36, 56, __ (a) 71 b) 76 c) 81 d) 86)
- Differences are 5, 10, 15, 20 — rising by 5 each time.
- The next difference is +25.
- 56 + 25 = 81.
Answer: 81 — option c)
Worked example
Find the next term: 7, 26, 63, 124, 215, __ (a) 342 b) 316 c) 330 d) 343)
- Each term is n^3 − 1: 2^3−1=7, 3^3−1=26, 4^3−1=63, 5^3−1=124, 6^3−1=215.
- The next is 7^3 − 1.
- 7^3 − 1 = 343 − 1 = 342.
Answer: 342 — option a)
Worked example
Find the next term: 4, 18, 48, 100, 180, __ (a) 270 b) 280 c) 294 d) 300)
- Take differences: 14, 30, 52, 80.
- Second differences rise by 6 each time, so the next first difference is 80 + 34 = 114.
- 180 + 114 = 294.
Answer: 294 — option c)
Worked example
Complete: AB, DE, HI, MN, __ (a) ST b) RS c) TU d) SR)
- Track the starting letters: A(1), D(4), H(8), M(13) — gaps 3, 4, 5.
- The next gap is +6, so 13 + 6 = 19 = S; the pair is the next two letters S, T.
- So the answer is ST.
Answer: ST — option a)
Worked example
Find the next term: 2, 3, 5, 7, 11, 13, __ (a) 15 b) 17 c) 19 d) 21)
- These are consecutive prime numbers.
- The prime after 13 is 17.
Answer: 17 — option b)
Worked example
Find the next term: 1, 2, 6, 24, 120, __ (a) 600 b) 720 c) 840 d) 150)
- Each term multiplies by 2, 3, 4, 5 in turn (factorials).
- The next multiplier is ×6.
- 120 × 6 = 720.
Answer: 720 — option b)
Worked example
1, 8, 27, 64, __ (a) 100 b) 121 c) 125 d) 144)
- These are perfect cubes: 1^3, 2^3, 3^3, 4^3.
- The next is 5^3 = 125.
Answer: 125 — option c)
Worked example
3, 7, 15, 31, __ (a) 47 b) 55 c) 63 d) 64)
- Each term is the previous ×2 + 1.
- 31 × 2 + 1 = 63.
Answer: 63 — option c)
⚠ Watch out
- Don't lock onto the first rule you see — verify it across all given terms.
- Letter series often skip in the alphabet; count positions, don't eyeball.
- Watch for a pattern that alternates between two operations.
Takeaways
- Always test differences, then ratios, then squares/cubes — in that order.
- Convert letters to position numbers before hunting for the step.
- If first differences don't settle, take the second differences.
- Verify a candidate rule against every term before committing.