Analytical Reasoning & Intelligence Tests — Chapter 2

Number & Letter Series, Sequences & Odd One Out

Why this chapter matters: Series is the second-largest reasoning family in Subject 12, behind only analogies, and the most mechanical marks in the section: no fact to memorise, only a rule to find. The script count behind this chapter (Section 10) finds 329 unique series/sequence/odd-one stems across 127 distinct files — 240 series, 21 sequence, 68 odd-one — and the family is still being asked right now: 8 files named 2024, 8 named 2025 and 8 named 2026. FPSC papers from 2001–2002 supply the classic worked patterns; PPSC papers supply the recent volume. Every worked example below is a real bank stem with its banked answer, and every rule is shown working on that stem — nothing is taught from a key whose pattern could not be reconstructed (see the honesty note in Section 10).

Status of this chapter (read first): this is AR-02, written 31 October 2026 in taxonomy order after AR-01. It replaces the short 15:12 PKT filing of the same chapter, whose front-matter still read “(script pending)” and which carried no worked bank examples; the counts here are recomputed by script at chapter time, as the Subject 12 structure page requires.

Exam Trend box

• 🔁 Most Repeated: Series/sequences/odd-one as a family — 329 unique stems across 127 distinct files (351 rows including duplicates). The family clears the 🔁 test twice over: far above 100 questions and far above 5 papers. Inside it, plain number series is the bulk (240 stems); odd-one-out is the second block (68 stems).
• ⭐ Must Know: Number series and odd-one-out — 🔁 and still asked in 2024, 2025 and 2026 papers by filename-year signal (8 / 8 / 8 files), including 2026 papers such as PPSC Assistant Disaster Management Department, PPSC Assistant Punjab Police (papers 3 and 4), PPSC PMS Clubbed Case (morning) and PPSC PMS SGAD, plus the CSS MPT 2026 mock and practice-set files. Expect a missing term, a wrong term, and a letter-series item in any reasoning section.
• 🆕 Newly Appeared / Recent Trend: none claimed. Series items run continuously from FPSC 2001–2002 papers to 2026 papers; their 2025–26 signal is prominence (a scored reasoning section in CSS MPT General Abilities), not first appearance.

1. The method — four steps, in order

1. Differences first. Subtract each term from the next. If the differences are constant, it is arithmetic; if the differences themselves have a pattern, take second differences.
2. Ratios second. If differences fail, divide each term by the previous one. A constant ratio is geometric; a ratio with a changing add/subtract is the ×k±c family.
3. Special numbers third. Check squares, cubes, primes, and n³−n / n²±n forms before inventing anything exotic.
4. Split fourth. If no single rule fits, split the series into odd-position and even-position streams — it is probably two interlaced series (see Section 6).

Never guess the rule from two terms: test it on every term given. In odd-one-out items the same discipline applies in reverse — find the rule the majority obey, and the term that breaks it is the answer.

2. Core notes — number series by pattern

2.1 Constant difference (arithmetic)

• Rule: add the same d each time. nth term = a₁ + (n−1)d.
• Worked example (banked): 120, 99, 80, 63, 48, ? — differences −21, −19, −17, −15, so the next difference is −13 and the answer is 35 (FPSC Assistant Director, Statistics Division, 2006).
• Worked example (banked): 4832, 5840, 6848, ? — differences +1008, +1008, so the answer is 7856 (FPSC Assistant Director, Establishment Division, 2001). Big numbers do not change the method.

2.2 Growing differences (second differences)

• Rule: the differences themselves increase by a constant step.
• Worked example (banked): 1, 5, 13, 25, 41, ? — differences 4, 8, 12, 16, so the next difference is 20 and the answer is 61 (FPSC Assistant Director, Minorities & Religious Division, 2011).
• Worked example (banked): 2, 2, 5, 13, 28, ? — differences 0, 3, 8, 15; second differences 3, 5, 7, so the next difference is 24 and the answer is 52 (FPSC Assistant Director, Ports & Shipping, 2002).

2.3 Multiply-and-adjust (×k±c)

• Rule: each term = previous × k, plus or minus a small number that itself follows a pattern.
• Worked example (banked): 6, 13, 28, 59, ? — ×2+1, ×2+2, ×2+3, so ×2+4 gives 122 (FPSC Customs Inspector, 2008).
• Worked example (banked): 3, 7, 15, ?, 63, 127 — ×2+1 throughout, so the missing term is 31 (FPSC Assistant Director, Minorities & Religious Division, 2011).
• Worked example (banked): 3, 7, 23, 95, ? — ×2+1, ×3+2, ×4+3, so ×5+4 gives 479 (FPSC Assistant Director, FIA, 2002). The multiplier itself is counting up — the trap is stopping at ×4+3 and answering 383.

2.4 Squares, cubes and special forms

• Rule: recognise the form before computing — squares of primes, odd squares, n³−n, and n²+1 chains are all bank favourites.
• Worked example (banked): 4, 9, 25, ?, 121, 169, 289, 361 — squares of the primes 2, 3, 5, 7, 11, 13, 17, 19, so the missing term is 49 (FPSC Assistant Director, FIA, 2002).
• Worked example (banked): 1, 9, 25, 49, 81, ? — odd squares 1², 3², 5², 7², 9², so the next is 11² = 121 (FPSC Assistant Director, Ports & Shipping, 2002).
• Worked example (banked): 24, 60, 120, 210, ? — n³−n for n = 3, 4, 5, 6, so n = 7 gives 343 − 7 = 336 (FPSC Inspector, ASF, 2009).
• Worked example (banked): 3, 12, 27, 48, 75, 108, ? — 3×1², 3×2², 3×3² … 3×6², so 3×7² = 147 (FPSC Assistant Director, FIA, 2002). Differences (9, 15, 21, 27, 33, 39) give the same answer — two routes, one answer.
• Worked example (banked): 3, 10, 101, ? — each term = previous²
  • 1 (3²+1 = 10, 10²+1 = 101), so 101²+1 = 10202 (FPSC Assistant Director, Statistics Division, 2006).

2.5 Add-the-previous-two (Fibonacci-type)

• Rule: each term = sum of the two before it (the starting pair need not be 1, 1).
• Worked example (banked): 5, 2, 7, 9, 16, 25, ? — 5+2 = 7, 2+7 = 9, 7+9 = 16, 9+16 = 25, so 16+25 = 41 (FPSC Assistant Director, FIA, 2002).

2.6 Alternating multiply / divide

• Rule: the operation alternates, and the number you multiply or divide by is itself the previous result.
• Worked example (banked): 6, 18, 3, 21, 7, 56, ? — 6×3 = 18, 18÷6 = 3, 3×7 = 21, 21÷3 = 7, 7×8 = 56, so 56÷7 = 8 (FPSC Assistant Director, FIA, 2002). Read the operations, not just the numbers: the multipliers/divisors are 3, 6, 7, 3, 8, 7 in pairs.

3. Interlaced (double) series — split before you solve

• Rule: odd positions form one series, even positions another. A single-rule attempt will always fail here; that failure is the signal to split.
• Worked example (banked): 20, 20, 19, 16, 17, 13, 14, 11, ?, ? — odd positions 20, 19, 17, 14, 10 (differences −1, −2, −3, −4); even positions 20, 16, 13, 11, 10 (differences −4, −3, −2, −1). Answer: 10, 10 (FPSC Customs Inspector, 2008).
• Worked example (banked): 2, 15, 4, 12, 6, 7, ?, ? — odd positions 2, 4, 6, 8; even positions 15, 12, 7, 0 (differences −3, −5, −7). Answer: 8, 0 (FPSC Assistant Director, Establishment Division, 2001).

4. Letter series

• Rule: convert letters to positions (A = 1 … Z = 26), find the numeric rule, convert back. Wrap around: after Z comes A.
• Worked example (banked): AB, DEF, HIJK, ?, STUVWX — groups of 2, 3, 4, 5, 6 letters, skipping one letter between groups (AB, skip C, DEF, skip G, HIJK, skip L, …), so the missing group is MNOPQ (FPSC Assistant Director, Ports & Shipping, 2002).
• Worked example (banked): A, C, F, J, O, ? — positions 1, 3, 6, 10, 15; gaps +2, +3, +4, +5, so the next gap is +6 and the answer is position 21 = U (CSS MPT General Abilities practice set). The gaps are triangular numbers in disguise.

5. Odd one out — find the property, not the odd look

• Rule: name one property the majority share (all cubes, all squares, all even, all prime, all multiples of 8). The term lacking it is the answer. “Looks different” is not a property.
• Worked examples (all banked, values as banked):
  • 8, 27, 64, 100, 125, 216, 343 — all the rest are cubes (2³, 3³, 4³, 5³, 6³, 7³); 100 is not a cube (PPSC papers 2019–2020).
  • 1, 4, 9, 16, 26, 36 — all the rest are squares; 26 should be 25 (FPSC Assistant Director, FIA, 2015).
  • 0, 1, 3, 6, 10, 15, 21, 28, 37, 45 — triangular numbers (differences 1, 2, 3, 4, 5, 6, 7, 8, 9); after 28 comes 36, not 37 (FPSC Inspector, ASF, 2009).
  • 8, 16, 24, 34, 40, 48 — all the rest are multiples of 8 (FPSC Assistant Security Officer, ASF, 2014).
  • 10, 14, 16, 18, 21, 24, 26 — all the rest are even (subject General Ability odd-man-out set).
  • 41, 43, 47, 53, 61, 71, 73, 81 — all the rest are prime; 81 = 9×9 (subject General Ability odd-man-out set).
  • 16, 25, 36, 72, 144, 196, 225 — all the rest are squares; 72 is not a square (subject General Ability odd-man-out set).
• Word odd-one-out works the same way (one city among countries, one season among months, one UN body among others in the banks), but those stems’ banked answers are option letters (B, D) with the options themselves not preserved in the bank files — so they are counted in the chapter total and not taught as worked examples here. A letter without its options is not a teachable answer.

6. Master table — pattern → test → example

7. Rapid-fire one-liners

• Differences first, ratios second, special numbers third, split fourth — in that order, every time.
• Arithmetic nth term = a₁ + (n−1)d; geometric nth term = a₁ × rⁿ⁻¹.
• Second differences constant ⇒ quadratic form (n², n³−n family).
• In ×k±c, watch whether k itself is counting (×2, ×3, ×4 …).
• Letter series: write A = 1 … Z = 26 above the letters before doing anything else.
• Odd one out: state the shared property in words before picking — “all cubes”, “all even”, “all multiples of 8”.
• A series with two answers (…, ?, ?) is almost always interlaced.

8. Most-asked facts

• 🔁 Difference and ×k±c number series dominate the 240 series stems; the FPSC 2001–2002 Assistant Director papers alone supply most of the classic patterns taught in Sections 2.1–2.6.
• 🔁 Odd-one-out by property (cubes, squares, primes, parity, multiples) is the second block at 68 stems — cheaper marks than series, because one property check answers the item.
• ⭐ Letter series stays in the must-know core: the group-length pattern (AB, DEF, HIJK, MNOPQ) and the growing-gap pattern (A, C, F, J, O, U) are both still the shapes recent practice sets use.
• The 21 “sequence” stems are the same skill under a different topic label (notably the CSS MPT General Abilities practice set) — learn one method, not two vocabularies.

9. Traps & confusions

• Interlaced mistaken for broken. If no single rule fits all terms, do not force one — split into odd/even streams (Section 3). Forcing a rule onto an interlaced series is the commonest error.
• Letter answers without options. Several bank odd-one stems bank only “B” or “D”. Those are counted, never taught: without the options, the letter proves nothing.
• Ambiguous stems held, not taught. Three bank stems were checked and deliberately left out of the worked examples because no rule reconstructs their banked key from the stem alone: the “13, 32, 24, 43, 35, 7, 46, 65, 57, 76” stem (key 54), the “12, 20, 32, 42, 56, 72, 90” stem (key 32), and the “3, 5, 9, 11, 14, 17, 21” odd-one stem (key 14). They stay in the count; they do not become teaching examples.
• Series vs analogy overlap. The n:(n²−1) pattern (2:3, 3:8, 4:15) belongs to AR-01 number analogies, not to this chapter — it is a relation between two numbers, not a sequence. Do not double-count it here.
• “Sequence” vs “series”. The banks use both labels for the same missing-term skill. The chapter count merges them (240 + 21); the student learns one method.

10. Memory tricks

• “D-R-S-S”: Differences, Ratios, Specials, Split — the four steps of Section 1 in order. Say the letters before touching the numbers.
• Squares to 20, cubes to 10, primes to 50 — recite them once before a reasoning section; half the odd-one items fall to that list alone.
• Alphabet strip: write 1–26 under A–Z in the margin for every letter item; never do letter gaps in your head.
• Interlace check: if the 1st, 3rd and 5th terms already look like their own series, you are done splitting — solve the two halves separately.

11. Visuals (specs only — site build renders these)

• Flow diagram: the four-step method (Section 1) as a decision tree (differences? → ratios? → specials? → split?).
• Table: squares 1²–20², cubes 1³–10³, primes to 50 — one reference card placed after Section 5.
• Worked-example panels: each Section 2 example with its differences / ratios annotated under the terms, so the rule is visible at a glance.

12. Practice this topic

Pull a timed set from the verified banks using the practice query in the front-matter (subject = analytical-reasoning, topic tags Series / Sequence / Odd Man Out / Odd One Out): 329 unique stems across 127 distinct files. Book mode: ← Previous: Analogies (AR-01) · Next: Coding-Decoding (AR-03) →

Sources & count (Section 10 honesty note). Count by script hidden_files/goldmine-notes-count-ar02.py (counts JSON hidden_files/goldmine-notes-ar02-counts.json), same bank dirs and topic-based classification as the Subject 12 structure count: 583 files / 56,388 questions scanned; 351 rows including duplicates; 329 unique stems; 127 distinct files (53 PPSC, 36 FPSC, 24 archive “paper-” files, 6 CSS, 6 subject-set, 2 NTS, by filename prefix); sub-families series 240 / sequence 21 / odd-one 68; filename-year files 2024: 8, 2025: 8, 2026: 8. The scanned totals are slightly higher than AR-01’s (579 files / 56,134 questions) because bank files were added between the two runs — the chapter count is the one recomputed at this chapter’s writing time, not carried forward. Classification is topic-based only, so untagged reasoning items inside general papers are not counted: 329 is a floor, stated honestly, not an estimate. Banked answers taught here were each verified against their stem by hand in this run; the three ambiguous stems and the letter-only odd-one stems named in Section 9 are counted but not taught. No fact in this chapter comes from memory; methods are standard procedural types, and every worked example is a bank stem with its banked answer.