5.2 Alphabetical & Number Series

Why Series Matter

Series items are the bread-and-butter of the Reasoning section: expect 5–7 of the 30 reasoning questions. They look intimidating because a long string of numbers or letters fills the line, but the underlying rules are short. The skill is differencing — turning a sequence into the gaps between its terms, which expose the rule.

Number Series: The Difference Method

Given a series 2, 6, 12, 20, 30, ?:

1. First-level differences: 6−2=4, 12−6=6, 20−12=8, 30−20=10 → 4, 6, 8, 10.
2. Second-level differences: 6−4=2, 8−6=2, 10−8=2 → constant 2.
3. So the first-level differences grow by +2 each time. The next difference is 10+2=12, and the next term is 30+12 = 42.

This is the classic n(n+1) pattern (2=1×2, 6=2×3, 12=3×4, …), but you do not need to recognise the closed form — the difference table solves it mechanically.

Worked Example — Geometric Series

3, 6, 12, 24, 48, ?

• First-level differences: 3, 6, 12, 24 — themselves doubling. That is the signature of a geometric series with ratio 2.
• Next term = 48 × 2 = 96.

Worked Example — Mixed Operation

1, 2, 6, 24, 120, ?

• 1 × 1 = 1, then 1 × 2 = 2, then 2 × 3 = 6, then 6 × 4 = 24, then 24 × 5 = 120.
• Rule: multiply by successive integers 1, 2, 3, 4, 5, … — this is the factorial pattern.
• Next term = 120 × 6 = 720.

Worked Example — Squares with an Offset

2, 5, 10, 17, 26, ?

• Compare to squares: 1, 4, 9, 16, 25 — each term is square + 1. So 1² + 1 = 2, 2² + 1 = 5, 3² + 1 = 10, 4² + 1 = 17, 5² + 1 = 26.
• Next term = 6² + 1 = 36 + 1 = 37.

This is the n² + 1 pattern — a common RRB variant. Once you spot that the differences (3, 5, 7, 9) are consecutive odd numbers, the underlying square structure is obvious. Any series whose first-level differences are consecutive odd numbers is built on squares plus or minus a small constant.

Worked Example — Cubes

1, 8, 27, 64, ?

• 1 = 1³, 8 = 2³, 27 = 3³, 64 = 4³.
• Next term = 5³ = 125.

Cube series are easy to spot once you memorise the first five cubes (1, 8, 27, 64, 125). If the differences grow very fast — much faster than an arithmetic pattern — check cubes before trying anything else.

Worked Example — Alternating Two Series

2, 5, 4, 10, 6, 15, ?

• Odd positions: 2, 4, 6, … (+2). Even positions: 5, 10, 15, … (+5).
• The next term is at an odd position (7th), so it is 6 + 2 = 8.

When a series looks irregular, split it into odd-position and even-position subsequences before trying anything else. Most 'chaotic' series on RRB are two clean series interleaved.

Worked Example — Difference-of-Differences

3, 7, 14, 25, 41, ?

• First differences: 4, 7, 11, 16.
• Second differences: 3, 4, 5 — increasing by 1 each time.
• Next second difference = 6, so next first difference = 16 + 6 = 22, and next term = 41 + 22 = 63.

This is a second-order arithmetic series. When the first differences are not constant, compute the second differences; if those are arithmetic, you have a second-order series and the next term follows mechanically.

Common Series Patterns at a Glance

Alphabetical Series

Letters are numbers in disguise. Convert each letter to its position (A=1, …, Z=26) and treat the sequence as a number series.

Worked example. A, C, F, J, O, ?

• Positions: 1, 3, 6, 10, 15.
• Differences: +2, +3, +4, +5 — the skip grows by 1 each time.
• Next skip = +6, so next position = 15 + 6 = 21 = U.

Worked example with backward skip. Z, X, U, Q, L, ?

• Positions: 26, 24, 21, 17, 12.
• Differences: −2, −3, −4, −5 — the backward skip grows by 1 each time.
• Next skip = −6, so next position = 12 − 6 = 6 = F.

Missing-Term vs Wrong-Term

• Missing term: The series has a ? in one position. Solve as above and substitute.
• Wrong term: Every term obeys the rule except one. Compute the differences across the entire series; the position where the difference breaks is where the wrong term sits. Then ask: what should be there under the rule? The number actually printed is the wrong term.

Worked example (wrong term). 2, 5, 10, 17, 26, 38, 50

• Differences: 3, 5, 7, 9, 12, 12.
• The first four differences (3, 5, 7, 9) follow +2 each time. The next should be 11, giving 26 + 11 = 37, not 38. So 38 is the wrong term; it should be 37.

Exam Traps

Trap 1 — Multiple plausible rules. The series 2, 4, 8, ? could be powers of 2 (→16) or n + 2, n × 2, … alternation (→10). Look at the available options: only one will be present.

Trap 2 — Alternating series. 1, 4, 3, 6, 5, 8, ? looks chaotic until you split it: odd positions 1, 3, 5, … (+2) and even positions 4, 6, 8, … (+2). The next term (odd position) = 5 + 2 = 7.

Trap 3 — Series that mix operations. 1, 3, 7, 15, 31, ? — differences 2, 4, 8, 16 — itself a doubling series, so the next difference is 32 and the next term is 31 + 32 = 63 (this is 2ⁿ − 1).

Speed Tips

• Write differences under the series as soon as you read it; the pattern usually jumps out.
• If the first-level differences are not arithmetic, immediately try multiplying (geometric) or computing second-level differences.
• Keep the eight patterns from the chart in working memory — most RRB series are one of them with a small twist.