3.1 Number Series

3.1 Number Series

Quick Answer: Number series items show a sequence and ask for the next or missing term. Identify the rule — arithmetic, geometric, second-level difference, squares/cubes, alternating, or Fibonacci-style — then apply it. Start by computing differences; they reveal most patterns within seconds.

The verbal intelligence section of the PAF GD Pilot Initial Test accounts for roughly 25% of total marks and is deliberately fast-paced: you get about 30 seconds per item. Number series questions are a staple — a sequence of 4–6 numbers appears and you must select the next term or fill a blank from four options.

Step-by-Step Method

Work through these checks in order — most items yield by step 3:

1. Compute first-level differences between consecutive terms. If constant → arithmetic series (add the same value each step).
2. Compute the ratio between consecutive terms. If constant → geometric series (multiply by the same value each step).
3. Compute second-level differences (differences of the differences). If those are constant, the series follows a quadratic rule of the form an² + bn + c.
4. Check squares and cubes — n², n³, n² ± k, n³ ± k. A constant offset from a perfect square is extremely common.
5. Split into two interleaved series if no single rule fits — odd-indexed terms follow one rule, even-indexed terms another.
6. Consider Fibonacci-style rules where each term is the sum (or difference) of the two preceding terms.

Common Series Types

Worked Example 1 — Decreasing Differences

Sequence: 1, 0.8, 0.65, 0.55, ?

Step 1 — compute differences (in magnitude):

• 1 − 0.8 = 0.20
• 0.8 − 0.65 = 0.15
• 0.65 − 0.55 = 0.10

Step 2 — inspect the differences: 0.20, 0.15, 0.10 — they decrease by 0.05 each step. The next decrement should be 0.05, giving 0.55 − 0.05 = 0.50.

Candidates have reported the keyed answer as 0.45, which implies the decrement plateaus at 0.10 (0.55 − 0.10 = 0.45). This is a useful reminder: when your calculated answer is not among the options, pick the closest one and re-check whether the pattern plateaus or shifts. Always test your rule against every supplied term, not just the last one.

Worked Example 2 — Second-Level Differences

Sequence: 2, 6, 12, 20, 30, ?

• First differences: 4, 6, 8, 10
• Second differences: 2, 2, 2 (constant)

Constant second differences signal a quadratic rule. The first differences increase by 2 each time, so the next first difference is 12, and the next term is 30 + 12 = 42. (This is the series n² + n.)

Worked Example 3 — Alternating Series

Sequence: 1, 10, 3, 20, 5, ?

No single rule fits all six terms. Split by position:

• Odd positions: 1, 3, 5 → arithmetic, +2 → next odd term is 7
• Even positions: 10, 20, ? → geometric, ×2 → next even term is 40

Since the blank is at position 6 (even), the answer is 40.

Traps to Avoid

• Two interleaved rules — the single most common trap. If differences look chaotic, split the series into odd and even positions before giving up.
• Decimal shifts — a series that looks like 2, 1, 0.5, 0.25 is geometric with ratio 0.5, not arithmetic.
• Off-by-one in position — squares indexed from 0 vs from 1 give different values (0²=0, 1²=1 vs 1²=1, 2²=4). Match the indexing to the first term.
• Negative ratios — a geometric series can alternate signs: 4, −8, 16, −32, 64 (ratio −2).
• Squares ± k masquerading as arithmetic — 3, 6, 11, 18, 27 has first differences 3, 5, 7, 9 (not constant), so it is NOT arithmetic; it is n² + 2.

Speed Tips

• Train your eye to recognise n² (1, 4, 9, 16, 25) and n³ (1, 8, 27, 64) instantly — recognising these sequences by sight saves 10–15 seconds.
• For alternating series, write odd-position terms and even-position terms on separate lines; the rule usually jumps out.
• If the first three checks fail, look at the answer choices — they often reveal the rule (e.g., if options are 36, 38, 40, 42 for a near-square series, squares ± k is likely).