Number and letter series for AFCAT: a checking order that finds the rule

A series question shows a few terms that follow a rule and asks for the next one, a missing one or a wrong one. The rule is almost always one of a small set: a steady difference, a growing difference, a multiplication, squares or cubes, or two series woven together.

The skill is not cleverness but order. If you run the same checks in the same sequence every time, most series fall within 30 to 45 seconds. Guessing at patterns wastes far more time.

The order of checks

1. Differences between terms. Constant differences mean an arithmetic series.
2. Differences of the differences. If the first differences change steadily, the second differences reveal how.
3. Ratios. If the terms grow fast, try dividing each term by the one before.
4. Multiply and add. Rules like × 2 + 1 or × 3 − 2 show up as nearly constant ratios.
5. Squares, cubes and primes. Compare each term with n², n³ or the list of primes, including ± 1.
6. Alternating series. If nothing fits, split the terms into odd and even positions.

Why this order: each check is quicker than the next, and each catches the most common rules before the rarer ones. Differences reveal slow, steady growth. Ratios reveal fast growth. Squares and cubes explain differences that are themselves odd numbers or growing steadily. Alternating series are the fallback when the terms jump up and down.

Why a rule must be checked against all terms

Take 1, 2, 4, ? with only three terms. Doubling gives 8. Differences 1, 2, then 3 give 7. Both rules are valid, which is why well-set questions give five or six terms. With more terms, only one rule survives. If two rules still fit all the given terms, check whether they give the same next term. They usually do, as in Example 1 below.

Number series: worked examples

Example 1 (growing difference). 2, 6, 12, 20, 30, ?

• Differences: 4, 6, 8, 10. They rise by 2, so the next difference is 12.
• 30 + 12 = 42.
• The terms are also 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, so the next is 6 × 7 = 42. Both descriptions are the same rule.

Example 2 (doubling difference). 3, 5, 9, 17, 33, ?

• Differences: 2, 4, 8, 16. Each doubles, so the next is 32.
• 33 + 32 = 65. As a check, each term is (previous × 2) − 1: 33 × 2 − 1 = 65.

Example 3 (alternating). 1, 10, 3, 20, 5, 30, ?

• The differences jump up and down, so split by position.
• Odd positions: 1, 3, 5, … Even positions: 10, 20, 30, …
• The seventh term is in the odd-position series: 7.

Example 4 (wrong term). Find the wrong term: 2, 5, 10, 17, 26, 38, 50.

• Differences: 3, 5, 7, 9, 12, 12. The odd numbers break at 12.
• Compare with n² + 1: 2, 5, 10, 17, 26, 37, 50. The sixth term should be 37.
• The wrong term is 38.

Letter series

Convert letters to positions: A = 1, B = 2, … Z = 26. Learn the anchors E = 5, J = 10, O = 15, T = 20, Y = 25, and the reverse position (A is 26th from the end, Z is 1st). If a series runs past Z, wrap round to A.

Example 5 (growing gap). A, C, F, J, O, ?

• Positions: 1, 3, 6, 10, 15. Gaps: +2, +3, +4, +5. Next gap +6.
• 15 + 6 = 21 = U.

Example 6 (letter groups). BDF, HJL, NPR, ?

• Inside each group, letters go up by 2: B, D, F.
• Each group starts 2 after the previous group ends: F → H, L → N.
• R → T, then T, V, X: TVX.

Example 7 (two directions). AZ, BY, CX, ?

• First letters go forward: A, B, C → D. Second letters go backward: Z, Y, X → W.
• Answer: DW. Each pair is also an opposite pair (positions add to 27).

Types at a glance

Practice set

1. 5, 10, 20, 40, ?
2. B, E, H, K, ?
3. 2, 3, 5, 8, 12, ?
4. 7, 12, 22, 37, 57, ?
5. 0, 7, 26, 63, 124, ?
6. 3, 7, 16, 35, 74, ?
7. Z, X, U, Q, L, ?
8. Find the wrong term: 1, 4, 9, 16, 24, 36, 49.

Answers:

1. 80. Each term doubles.
2. N. Positions 2, 5, 8, 11, going up by 3, so 14 = N.
3. 17. Differences 1, 2, 3, 4, so the next is 5. (It is not the sum of the previous two, since 5 + 8 = 13, not 12.)
4. 82. Differences 5, 10, 15, 20, so the next is 25.
5. 215. Each term is n³ − 1: 1 − 1, 8 − 1, 27 − 1, 64 − 1, 125 − 1, then 216 − 1.
6. 153. Each term is double the previous plus 1, 2, 3, 4, then 5: 74 × 2 + 5 = 153. The differences (4, 9, 19, 39, 79) confirm it, since each is double the one before plus 1.
7. F. Positions 26, 24, 21, 17, 12. Gaps −2, −3, −4, −5, so the next is −6: 12 − 6 = 6 = F.
8. 24. The terms are squares: 1, 4, 9, 16, 25, 36, 49. The fifth should be 25.

What to do next

• Learn squares to 30, cubes to 15, primes to 100 and powers of 2 to 1,024. Most series rules use them.
• Practise the six checks in order on 20 series a day for a week, timing each one.
• Move on to coding–decoding, which uses the same letter positions, and to figure series, which applies the same logic to shapes.