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Series (Number & Letter)

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high importance~4 Q in Tier 16 formulas⚡ 10 shortcuts6 subtopics
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Subtopic 2 of 6·← How every series is cracked·Letter series →

Number series

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The CGL number-series rule bank, in frequency order:

  1. Linear recurrence: a(n+1) = k·a(n) ± r — e.g. 5, 11, 23, 47, 95 is ×2 + 1.
  2. Second-difference (n² family): 2, 6, 12, 20, 30 is n² + n; 3, 8, 15, 24 is n² − 1.
  3. Cube/jump differences: differences are 1, 8, 27, 64 (cubes) or 1, 2, 4, 8 (powers of 2).
  4. Rising multipliers: ×2, ×3, ×4 — e.g. 7, 14, 42, 168, 840.
  5. Alternating/interleaved: two mini-series on odd and even positions.
  6. Fibonacci style: each term = sum of previous two.

Uniqueness discipline: after guessing the rule, check it reproduces every given term — not just the last. If two rules both fit, the exam keeps the second rule's answer out of the options; if your answer is missing, drop the rule, don't force it.

📖 Notes🧩 8 question types⚡ Formulas🎯 26 practice Q

Detailed notes

What this lesson covers

Number series is the most common series type in competitive exams. Almost every question uses one of eight families. Learn the families and every question becomes a quick check against a short list.

1. Constant or growing difference

Add the same number every time: 7, 12, 17, 22 (+5). This is called an arithmetic progression (AP). In the growing type, the amount added itself grows: 11, 13, 17, 23, 31 → add 2, 4, 6, 8, so next add 10 → 41. It can also fall: 120, 119, 116, 111 (subtract 1, 3, 5).

2. Multiply, then add or subtract (×k ± r)

Each term is the previous term multiplied by a fixed number, sometimes with a small add or subtract. 3, 6, 12, 24 is ×2. 3, 7, 15, 31 is ×2 + 1. 4, 10, 28, 82 is ×3 − 2.

Test: divide a big term by the one before. If the result is near 2 or 3, try ×2 or ×3 and see what small number is left over. That left-over is r.

3. Squares and cubes

Know squares up to 30 and cubes up to 15. Look for terms just above or below them:

  • 3, 8, 15, 24, 35 = 2²−1, 3²−1, 4²−1 ...
  • 2, 9, 28, 65 = 1³+1, 2³+1, 3³+1, 4³+1
  • 2, 6, 12, 20, 30 = 1×2, 2×3, 3×4 ... (n × (n+1))

4. Two-level difference

The first difference row is not constant, but the second row follows a clear rule.

Example: 2, 5, 11, 21, 36 → differences 3, 6, 10, 15 → second differences 3, 4, 5. Next second difference is 6, so the next difference is 21 and the next term is 57.

5. Differences that are famous numbers

The difference row itself may be squares (1, 4, 9, 16), cubes (1, 8, 27, 64), primes (2, 3, 5, 7, 11) or powers of 2 (1, 2, 4, 8).

Example: 10, 11, 13, 17, 25 → differences 1, 2, 4, 8 → next 16 → 41.

6. Alternate (interleaved) series

Two separate series are mixed. Terms 1, 3, 5, 7 form one series; terms 2, 4, 6, 8 form another.

Example: 3, 50, 6, 45, 12, 40, 24, ? → odd places 3, 6, 12, 24 (×2); even places 50, 45, 40 (−5). The 8th term is an even place → 35. Suspect this when the series goes up and down.

7. Mixed operations

The operation changes in a fixed way:

  • ×1+1, ×2+2, ×3+3 ...: 2, 3, 8, 27, 112 → next 112×5+5 = 565.
  • Two operations taking turns (+1, ×2, +1, ×2): 5, 6, 12, 13, 26, 27 → 54.
  • Changing signs (+2, −3, +4, −5): 10, 12, 9, 13, 8 → 14.

8. Fibonacci type

Each term is the sum of the previous two (2, 3, 5, 8, 13, 21) or of the previous three (1, 2, 3, 6, 11, 20, 37). Spot it when the difference row looks like the series itself.

How to choose quickly

  1. Growth is slow → difference ladder (families 1, 4, 5).
  2. Growth is fast → ratio test (families 2, 7).
  3. Terms go up and down → alternate series (family 6).
  4. Terms near squares or cubes → family 3.
  5. Each term is close to the sum of the two before it → Fibonacci (family 8).

Quick revision

FamilyExampleNext
Constant difference7, 12, 17, 2227
Growing difference11, 13, 17, 23, 3141
×k ± r3, 7, 15, 3163
Squares/cubes3, 8, 15, 2435
Two-level difference2, 5, 11, 21, 3657
Famous differences10, 11, 13, 17, 2541
Alternate3, 50, 6, 45, 12, 4024
Mixed operations2, 3, 8, 27, 112565
Fibonacci2, 3, 5, 8, 1321

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Constant or growing difference (AP and near-AP)very common2 practice Q
How to spot it:

Terms grow slowly and steadily; the gaps are equal or rise/fall by the same amount.

an+1=an+d,d constant or rising evenlya_{n+1} = a_n + d,\quad d \text{ constant or rising evenly}
  1. Write the differences.
  2. If they are equal, add the same number again.
  3. If they change by a fixed amount (2, 4, 6, 8), continue that row.

Why it works: an adding rule shows itself directly in the difference row.

Example: 4, 6, 10, 16, 24, ?

Differences: 2, 4, 6, 8 → next 10. 24 + 10 = 34.

Type 2: Multiply and add/subtract (×k ± r)very common2 practice Q
How to spot it:

Terms roughly double or triple each time; the ratio is almost (but not exactly) constant.

an+1=k an±ra_{n+1} = k\,a_n \pm r
  1. Divide a term by the previous one to guess k (2, 3, 4 ...).
  2. Compute k × previous and see the small left-over r.
  3. Confirm the same k and r for every step, then apply.

Why it works: a fixed multiply-then-add rule makes the ratio almost constant, and the left-over is the same each time.

Example: 2, 5, 14, 41, ?

2×3 − 1 = 5, 5×3 − 1 = 14, 14×3 − 1 = 41. Next: 41×3 − 1 = 122.

Type 3: Squares, cubes and n(n+1) based termsvery common3 practice Q
How to spot it:

Terms sit on or just beside squares (4, 9, 16, 25) or cubes (8, 27, 64), or look like 2, 6, 12, 20.

an=n2±k,  n3±k,  n(n+1)a_n = n^2 \pm k,\ \ n^3 \pm k,\ \ n(n+1)
  1. Compare every term with the nearest square or cube.
  2. Find the fixed adjustment (−1, +1, +2) or the n(n+1) form.
  3. Use the next n.

Why it works: squares and cubes grow in a special way, so terms near them are easy to recognise once you know the tables.

Example: 0, 3, 8, 15, 24, ?

Terms are 1²−1, 2²−1, 3²−1, 4²−1, 5²−1. Next: 6² − 1 = 35.

Type 4: Two-level difference (second differences follow a rule)common2 practice Q
How to spot it:

The first differences are not equal, but they themselves rise in a pattern (1, 3, 6, 10 ...).

Δ2an follows a simple rule\Delta^2 a_n \text{ follows a simple rule}
  1. Write the first difference row.
  2. Write the second difference row under it.
  3. Continue the second row, then the first row, then the series.

Why it works: each extra ladder row removes one layer of the rule until a plain pattern remains.

Example: 3, 4, 7, 13, 23, ?

First differences: 1, 3, 6, 10. Second: 2, 3, 4 → next 5. Next first difference = 10 + 5 = 15. 23 + 15 = 38.

Type 5: Alternate (interleaved) two-in-one seriescommon2 practice Q
How to spot it:

The series goes up and down, or big and small numbers take turns.

  1. Separate the terms in odd places (1st, 3rd, 5th ...) and even places (2nd, 4th ...).
  2. Solve each small series on its own.
  3. Decide whether the '?' is an odd or even place, and take the answer from that series.

Why it works: two independent rules are simply written one after the other.

Example: 1, 10, 2, 20, 3, 30, ?

Odd places: 1, 2, 3 → next 4. Even places: 10, 20, 30. The 7th term is an odd place → 4.

Type 6: Mixed operations (×1+1, ×2+2 ... or operations taking turns)common3 practice Q
How to spot it:

Terms grow faster and faster with no constant ratio; or operations clearly take turns (+, ×, +, ×).

an+1=n an+na_{n+1} = n\,a_n + n
  1. Try ×1+1, ×2+2, ×3+3 (or ×1−1, ×2−2 ...).
  2. Try two operations taking turns (+1 then ×2).
  3. Try gaps with changing signs (+2, −3, +4, −5).
  4. Continue the operation pattern for the next step.

Why it works: the operation itself forms a simple series, so the next operation is predictable.

Example: 1, 2, 6, 21, 88, ?

1×1+1 = 2, 2×2+2 = 6, 6×3+3 = 21, 21×4+4 = 88. Next: 88×5+5 = 445.

Type 7: Fibonacci type (sum of previous terms)occasional2 practice Q
How to spot it:

Each term is close to the sum of the two (or three) terms before it.

an+2=an+1+ana_{n+2} = a_{n+1} + a_n
  1. Add the previous two terms and compare with the next term.
  2. If that fails, add the previous three.
  3. Continue the same addition.

Why it works: such a series grows by feeding on itself, so its difference row copies the series.

Example: 1, 1, 2, 3, 5, 8, ?

Each term = sum of the previous two. 5 + 8 = 13.

Type 8: Differences are squares, cubes, primes or powers of 2common2 practice Q
How to spot it:

The difference row is itself a famous list: 1, 4, 9, 16 / 1, 8, 27 / 2, 3, 5, 7, 11 / 1, 2, 4, 8.

an+1=an+n2 (or n3, pn, 2n)a_{n+1} = a_n + n^2 \ (\text{or } n^3,\ p_n,\ 2^n)
  1. Write the differences.
  2. Match the row to squares, cubes, primes or powers of 2.
  3. Take the next member of that list and add it.

Why it works: the examiner simply adds a famous list term by term.

Example: 5, 6, 10, 19, 35, ?

Differences: 1, 4, 9, 16 (squares) → next 25. 35 + 25 = 60.

Formulas

AP / ladder
an=a1+(n−1)da_n = a_1 + (n-1)d

d = common difference from the ladder.

Recurrence xk plus r
an+1=k⋅an±ra_{n+1} = k \cdot a_n \pm r

Test k = 2, 3 first with r = ±1, ±2.

n-squared family
an=n2±k or n2+na_n = n^2 \pm k \text{ or } n^2 + n

Second difference 2 ⇒ n²; second difference 2k ⇒ k·n².

Geometric
an=a1⋅r n−1a_n = a_1 \cdot r^{\,n-1}

Constant ratio r; check ×2, ×3, ×1.5.

Shortcut tricks

⚡ Ratio test before exotic rules

Divide each term by the previous one. Constant ratio → GP. Ratio drifting by +1 each time → rising multipliers. Neither → back to the ladder.

Example: Q. 7, 14, 42, 168, ?

Sol. Ratios: 2, 3, 4 → next ratio 5. 168 × 5 = 840.

One division per gap; rising multipliers announce themselves immediately.

⚡ Interleaved split for jumpy series

If the series jumps up-down-up-down, write odd-position terms in one row and even-position terms below. Each row is a friendly AP/GP; extend both, then read the term that falls on '?'

Example: Q. 2, 9, 4, 13, 6, 17, ?

Sol. Row A (odd): 2, 4, 6 → 8. Row B (even): 9, 13, 17 → 21. '?' is 7th term → 8.

Splitting kills panic; each row is solvable in 3 seconds.

⚡ Cubes and powers hiding in the ladder

Memorise the signature ladders: 1, 8, 27, 64 (cubes), 1, 3, 5, 7 (odds), 2, 4, 8, 16 (powers of 2), 1, 4, 9, 16 (squares). When the first difference row matches a signature, the series is solved.

Example: Q. 12, 13, 21, 48, 112, ?

Sol. Differences: 1, 8, 27, 64 → cubes! Next difference 125. 112 + 125 = 237.

The ladder names the family — never subtract twice when the first row already looks famous.

Where students lose marks

  • Fitting ×2+1 from the first pair only and never verifying on the third term.

  • Missing mixed operators (×2+1 vs ×2) — check the residual r on every step.

  • In interleaved series, reading the answer from the wrong row (count positions carefully).

Practice sets — 26 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.