🧮 Aptitude & Reasoning · Chapter 8

Number series and pattern spotting

Every number series in a placement test comes from a list of about eight rules — check them in the right order and most series fall in under twenty seconds.

Study this lesson free in the app →

The basics

A number series is a list of numbers built by one rule; you find the rule and apply it to get the missing or next term.

The problem it solves: series test whether you can spot structure fast. They're 3–5 cheap marks in almost every placement paper — cheap only if you have a checking order instead of staring.

The mental model: look at how fast the terms grow, and let that pick the first test. Slow, steady growth → take the differences. Roughly doubling or tripling → test a ratio, often with a ±1. Numbers near 9, 16, 25 or 27, 64 → squares or cubes. Up-and-down jumps → two series interleaved. Growth that keeps accelerating → multipliers that grow (×1, ×2, ×3…).

Then confirm on every term, not just the first two.

Explained

5, 11, 23, 47, 95, ?

The slow way: differences 6, 12, 24, 48 — they double, so the next is 96, and 95 + 96 = 191.

The topper's way: each term is just over double the last. Test ×2 + 1: 5→11→23→47→95→191. Two seconds.

Same answer, but the second route came from one glance at how fast the numbers grow. That glance is the skill.

The orange pyramid at Crawford Market

A fruit seller at Crawford Market stacks oranges in a square pyramid. Top layer: 1. Next: a 2×2 square, 4. Then 9, then 16, then 25.

His nephew counts the total after each layer: 1, 5, 14, 30, 55.

Ask the nephew for the next total and he'd stare. Ask the seller and he'd say "add a 6×6 layer — 36 more — 91."

The seller knows the rule because he built the series. The differences — 4, 9, 16, 25 — are the layers. That's the trick for every series: take the differences and see if the thing that's being added each step is something you recognise.

The difference table

 series        7     10     16     25     37      ?
                 \  /   \  /   \  /   \  /   \  /
 1st diff         3      6      9     12     (15)
                   \    / \    / \    / \    /
 2nd diff            3      3      3      3

 2nd difference constant → next 1st diff = 12 + 3 = 15
 next term = 37 + 15 = 52

Work downwards until a row is constant, then build back up. If the first row is constant it's arithmetic; if the second is constant it's a squared rule (here 7 + 3·n(n−1)/2).

If no row settles after two levels, stop subtracting — switch to ratios or interleaving.

The checklist, in testing order

CUE (how it grows)          TEST                    EXAMPLE
steady, small growth        1st / 2nd differences   7,10,16,25,37 → 52
about ×2, ×3                ratio, then ×k ± c      5,11,23,47 → 95 (×2+1)
near squares / cubes        n² ± c, n³ ± c          3,8,15,24,35 → 48 (n²−1)
accelerating fast           ×1, ×2, ×3 … (± n)      1,2,6,21,88 → 445 (×n+n)
shrinking by big steps      ÷ 6, ÷ 5, ÷ 4 …         720,120,24,6,2 → 1
zig-zag up and down         odd / even positions    3,10,6,20,12,40 → 24
small, irregular            primes, Fibonacci        2,3,5,7,11 → 13

Know cold: squares to 30², cubes to 12³, powers of 2 to 2¹⁰, primes to 50.

The shortcut: compare each term with double the previous. If the gaps are +1, +1, +1… it's ×2 + 1; if they're +1, +2, +3… it's ×2 + n. This catches ×k ± c series without computing ratios.

When it fails: if the multiplier changes (×1, ×2, ×3), the gaps won't be regular — test the growing multiplier instead.

Solving steps

For wrong-number questions, step 4 is the whole question: the term that breaks the confirmed rule is the answer.

Five series, two minutes

(15 s) 3, 8, 15, 24, 35, ? — each is one less than a square (4, 9, 16, 25, 36). Next: 49 − 1 = 48.

(15 s) 720, 120, 24, 6, 2, ? — shrinking by big steps: ÷6, ÷5, ÷4, ÷3, so ÷2 → 1.

(30 s) 1, 2, 6, 21, 88, ? — accelerating. ×1+1, ×2+2, ×3+3, ×4+4 → 88 × 5 + 5 = 445.

(20 s) 3, 10, 6, 20, 12, 40, ? — zig-zag. Odd positions 3, 6, 12 (×2); even 10, 20, 40. The 7th term is odd-position → 24.

(30 s) Wrong number: 6, 12, 21, 33, 49, 66 — gaps 6, 9, 12, 16, 17. They should be 6, 9, 12, 15, 18, which gives 48 and then 66 again. So 49 is wrong.

Total: under two minutes, because each started with the growth glance, not with random subtraction.

The stuttering series

2, 3, 6, 7, 14, 15, ?

Traps examiners set

• Checking only the first two terms — 2, 6, 12, 20 starts like ×3 but it's n(n+1). Confirm on all terms
• Subtracting forever — if the second differences aren't constant, a third level is rarely the answer; switch to ratios
• Missing interleaving — up-down-up-down almost always means two series; split odd and even positions
• Wrong-number fixes — after you find the bad term, check the term after it still fits; otherwise you picked the wrong one
• Position of the missing term — in an interleaved series, decide whether the blank is odd or even before applying a rule
• Off-by-one in squares — is it n² − 1 starting from n = 2, or n² + 1 from n = 1? Test against the first term

Words to know

In two minutes

• A number series is built by one rule; your job is to find it fast and confirm it on every term.
• Let the growth pick the test: steady → differences, ×2/×3 → ratio or ×k ± c, near squares → n² ± c.
• If first differences aren't constant, try second differences once — then switch to ratios.
• Zig-zag series are two series interleaved: split odd and even positions.
• Fast acceleration usually means a growing multiplier: ×1 + 1, ×2 + 2, ×3 + 3 …
• For wrong-number questions, confirm the rule on every term; the one that breaks it is the answer.
• Memorise squares to 30, cubes to 12, powers of 2 and primes to 50 — recognition beats calculation.

Test yourself

There are 10 questions, 3 puzzles and 5 recall cards for this lesson in the app — the recall cards come back on later days, just before you would forget.