Number Series — What & Why?

A Number Series is an ordered sequence of numbers following a hidden mathematical pattern. Your task is to find the missing term or the next term.

Types tested in BPSC TRE:

• ›Arithmetic series — constant addition or subtraction
• ›Geometric series — constant multiplication or division
• ›Square/Cube series — powers of numbers
• ›Mixed/Two-step series — two operations alternating
• ›Prime number series — consecutive primes

Golden rule: Always check the difference between consecutive terms first — this reveals most patterns instantly.

Type 1 — Arithmetic Series

Each term increases or decreases by a constant difference (d).

Examples:

• ›+3: 2, 5, 8, 11, 14, ? — 17
• ›−4: 40, 36, 32, 28, ? — 24
• ›+1.5: 1, 2.5, 4, 5.5, ? — 7

Increasing difference:

• ›+1, +2, +3, +4… : 1, 2, 4, 7, 11, ? — 16
• ›+2, +4, +6, +8… : 2, 4, 8, 14, 22, ? — 32

Method: Subtract each term from the next — if differences are equal = arithmetic. If differences increase by a fixed amount = increasing-gap arithmetic.

Type 2 — Geometric Series

Each term is multiplied or divided by a constant ratio (r).

Examples:

• ›×2: 3, 6, 12, 24, 48, ? — 96
• ›×3: 2, 6, 18, 54, ? — 162
• ›÷2: 128, 64, 32, 16, ? — 8
• ›×0.5: 80, 40, 20, 10, ? — 5

Mixed multiply-add:

• ›×2+1: 1, 3, 7, 15, 31, ? — 63 (×2+1 each time)

Method: Divide each term by the previous — if ratio is constant = geometric series. If not, try ×n ± constant pattern.

Type 3 — Square & Cube Series

Perfect square series:

• ›n²: 1, 4, 9, 16, 25, 36, ? — 49 (7²)
• ›n²+1: 2, 5, 10, 17, 26, ? — 37 (6²+1)
• ›n²−n: 0, 2, 6, 12, 20, ? — 30 (n²−n)

Perfect cube series:

• ›n³: 1, 8, 27, 64, 125, ? — 216 (6³)
• ›n³+n: 2, 10, 30, 68, ? — 130 (n³+n)

Key squares to memorize: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225

Key cubes to memorize: 1,8,27,64,125,216,343,512

Type 4 — Mixed & Prime Series

Two-step alternating series:

• ›×2, +3 alternating: 1, 2, 5, 10, 13, 26, ? — 29 (+3)
• ›+5, ×2 alternating: 2, 7, 14, 19, 38, ? — 43 (+5)

Prime number series:

• ›Consecutive primes: 2, 3, 5, 7, 11, 13, ? — 17
• ›Prime + 1: 3, 4, 6, 8, 12, 14, ? — 18 (prime+1 pairs)

Odd one out in number series:

• ›Find the term that breaks the established pattern
• ›3, 6, 12, 24, 49, 96 — should be 48 (×2 each time) — 49 is wrong

Tip: For alternating series, check odd-positioned and even-positioned terms separately.

Points to Remember

• ›First step always — find differences between consecutive terms
• ›Constant difference = arithmetic series (+n or −n)
• ›Increasing difference = gaps grow by fixed amount (+1,+2,+3…)
• ›Constant ratio = geometric series (×n or ÷n)
• ›Squares: 1²–15² must be memorized for instant recognition
• ›Cubes: 1³–8³ must be memorized
• ›Prime series — 2,3,5,7,11,13,17,19,23,29,31,37…
• ›Alternating series — check odd and even positions separately
• ›×2+1 pattern — very common: 1,3,7,15,31,63…
• ›Exam tip: Students apply addition when the pattern is actually multiplication — always divide two consecutive terms to check for geometric ratio before assuming arithmetic.
• ›Exam tip: In square series like 2,5,10,17,26,? — students pick 36 (next square) instead of 37 (n²+1) — always check if a constant is added to each square.

Letter & Symbol Series — What & Why?

A series is an ordered sequence of letters, symbols, or both that follows a hidden pattern or rule.

Your task: Identify the pattern — predict the missing or next term.

Three types tested in BPSC TRE:

• ›Letter Series — based on alphabetical positions
• ›Symbol Series — repeating visual/sign patterns
• ›Mixed Series — letters + numbers + symbols combined

Why it matters: This topic tests logical pattern recognition — a core skill in every competitive exam.

Letter Series — Alphabetical Patterns

Foundation rule: A=1, B=2, C=3 … M=13, N=14 … Z=26

Common patterns:

• ›+n jump: A, C, E, G, ? — +2 each — I
• ›−n jump: Z, X, V, T, ? — −2 each — R
• ›Increasing jump: A, B, D, G, K, ? — +1,+2,+3,+4,+5 — P
• ›Reverse alphabet: Z=1, Y=2 … A=26 (used in opposite-end questions)
• ›Group patterns: ABD / CEG / FHJ — each group skips differently

Key tip: Write position numbers below letters to spot the arithmetic instantly.

Symbol Series — Cyclic Repeating Patterns

Symbols repeat in a fixed block of 2, 3, or 4.

Examples:

• ›Block of 2: ★ ☿ ★ ☿ ★ ? — ☿
• ›Block of 3: △ ○ □ △ ○ □ △ ? — ○
• ›Block of 4: ♦ ♣ ♥ ♠ ♦ ♣ ♥ ? — ♠

Method:

1. 1.Spot the repeating block (try blocks of 2, 3, 4)
2. 2.Find which position in the block the blank occupies
3. 3.Pick the matching symbol

Tip: Always verify with at least two full cycles before confirming your block.

Mixed Series — Letters + Numbers + Symbols

Each element in a mixed term follows its own independent rule.

Example: A1★, C3☿, E5★, G7☿, ?

• ›Letters: A→C→E→G→I (+2 each)
• ›Numbers: 1→3→5→7→9 (+2 each)
• ›Symbols: ★☿★☿→★ (alternating)
• ›Answer: I9★

Strategy:

1. 1.Separate the series into letter, number, symbol tracks
2. 2.Solve each track independently
3. 3.Recombine for the final answer

Never try to solve all three tracks at once — split first, then merge.

Odd One Out in a Series

Instead of finding the next term, find the term that breaks the pattern.

Example: B2, D4, F6, H9, J10

• ›Rule: letter +2, number +2
• ›B2→ D4→ F6→ H9✗ (should be H8) J10→
• ›Odd one out: H9

Step-by-step approach:

1. 1.Establish rule from first 3 terms
2. 2.Test each remaining term against the rule
3. 3.Mark the first violation as the answer
4. 4.Double-check: confirm all other terms satisfy the rule

Trap: Later terms may also seem wrong due to a cascading error — focus on the first breaking point.

Points to Remember

• ›A=1 to Z=26 — must memorize for quick solving
• ›+n / −n patterns — most common letter series type
• ›Increasing jump — gap increases by 1 each step (+1,+2,+3…)
• ›Symbol block — always 2–4 symbols; verify with 2 full cycles
• ›Mixed series — split into 3 independent tracks, then merge
• ›Odd one out — rule from first 3 terms, check each term after
• ›Reverse alphabet — Z=1 used in mirror/opposite type questions
• ›Group series — terms come in pairs or triplets, not individually
• ›Exam tip: In symbol series, students assume alternation (block of 2) when the block is actually 3 — always count 6+ symbols before deciding block size.
• ›Exam tip: In mixed series, students solve only letters and numbers, ignoring symbols — the symbol track is often what makes one option wrong; always check all three.