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Number Series

Competitive Mathematics · Number System

7 Part4 worked examples12 practice questions

A number series is a row of numbers built by some rule. A question asks for the next term, a missing term in the middle, or the one term that breaks the rule. This chapter teaches you to recognise every kind of rule — from plain adding and multiplying to squares, cubes, primes and woven series.

Every idea comes first in plain words, then solved examples, then questions for you (easy to hard). All the numbers are our own.

1

Reading a series

Start with gaps and ratios

The first way to open any series, and the commonest rules — adding or subtracting (AP) and multiplying or dividing (GP).

The order for finding the pattern

Five steps, every time

Check every series in the same order; most open up in the first two steps.

Do not panic when you see a series. Check in a fixed order; the patterns are always one of five or six kinds. When a pattern fits the first few terms, confirm it on the last ones.

The order of checks
1. Write the gaps2. Equal gaps? → AP3. Check the ratios → GP4. Gaps of gaps, or compare with squares and cubes5. Primes, woven or mixed
What you seeLikely patternExample
Equal gapsAdd / subtract series4, 11, 18, 25, 32
Equal ratiosMultiply / divide series3, 9, 27, 81, 243
Gaps follow a ruleEqual second gaps2, 5, 10, 17, 26
Numbers close to squares11, 18, 27, 38, 51
Numbers close to cubes7, 26, 63, 124, 215
Irregular but all primePrime series7, 11, 13, 17, 19
Behaviour alternatesTwo woven series5, 20, 9, 30, 13, 40
Example 1 · Basic
Find the next term: 7, 13, 19, 25, 31, ?
Solution, step by step
  1. 1Gaps between neighbours: 6, 6, 6, 6.
  2. 2Gaps of the gaps: 0, 0, 0.
  3. 3Pattern: each term is the previous term plus 6.
  4. 4The next term = 37.
Answer 37
Example 2 · Basic
Find the next term: 5, 10, 20, 40, 80, ?
Solution, step by step
  1. 1Gaps between neighbours: 5, 10, 20, 40.
  2. 2Ratios: 2, 2, 2, 2 — all equal.
  3. 3Pattern: each term is 2 times the previous term.
  4. 4The next term = 160.
Answer 160
BasicTry it yourself first
Find the next term of the series: 12, 21, 30, 39, 48, ?
BasicTry it yourself first
Find the next term of the series: 95, 88, 81, 74, 67, 60, ?
BasicTry it yourself first
Find the next term of the series: 2, 6, 18, 54, 162, ?
BasicTry it yourself first
Find the next term of the series: 7,776, 1,296, 216, 36, 6, ?

AP, GP and ×a+b

The three commonest rules

Equal gaps (AP), equal ratios (GP), and “multiply then add or subtract”.

AP: the same number is added (or subtracted) every time. With gap the th term is . GP: each term is a fixed multiple of the one before; with ratio the th term is . ×a+b: the previous term is multiplied by and is added or subtracted — gaps are not equal and ratios are not equal, but the gaps themselves form a GP.

The first is for an AP, the second for a GP.
first term
common gap
common ratio
Example 3 · Standard
Find the next term: 3, 7, 15, 31, 63, ?
Solution, step by step
  1. 1Gaps between neighbours: 4, 8, 16, 32.
  2. 2Gaps of the gaps: 4, 8, 16.
  3. 3Pattern: each term = previous × 2 + 1.
  4. 4The gaps 4, 8, 16, 32 double every time — so the rule is ×2 + 1.
  5. 5The next term = 127.
Answer 127
Example 4 · Standard
Find the next term: 4, 10, 28, 82, 244, ?
Solution, step by step
  1. 1Gaps between neighbours: 6, 18, 54, 162.
  2. 2Gaps of the gaps: 12, 36, 108.
  3. 3Pattern: each term = previous × 3 − 2.
  4. 4The next term = 730.
Answer 730
StandardTry it yourself first
Find the next term of the series: 2, 7, 22, 67, 202, ?
StandardTry it yourself first
Find the next term of the series: 5, 7, 11, 19, 35, ?
BasicTry it yourself first
Find the next term of the series: −8, 3, 14, 25, 36, ?
BasicTry it yourself first
Find the next term of the series: 1, 5, 25, 125, 625, ?
AdvancedTry it yourself first
Find the next term of the series: 9, −18, 36, −72, 144, ?
BasicTry it yourself first
Find the missing term: 14, 27, 40, ?, 66, 79, 92
StandardTry it yourself first
Find the missing term: 4, 12, 36, 108, ?, 972, 2,916
StandardTry it yourself first
Find the missing term: 6, 15, 33, 69, ?, 285, 573
2

Squares, cubes and gaps of gaps

Series near perfect squares and cubes

When numbers grow fast but ratios are not equal, look for squares, cubes or gaps of gaps.

Square and cube series

, , ,

Recognising perfect squares and cubes — and numbers just next to them.

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Gaps of the gaps

When the gaps are themselves a series

If the first gaps are not equal, look at the gaps of the gaps, or check whether the gaps are squares or primes.

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3

Special patterns

Primes · woven · mixed

When plain gaps or ratios fail, try primes, two woven series and mixed operations.

Prime series

Consecutive and with skips

When the numbers look irregular but are all prime.

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Woven series

Odd and even places run separately

When two rules take turns, look at the odd-place and even-place terms separately.

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Sum-of-previous and growing-multiplier series

Terms built from earlier terms

A series whose every term is the sum of the previous two, or whose multiplier keeps growing.

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4

Types of questions

Missing term and wrong term

The two main exam questions on series — finding a missing term and finding the wrong term — and their tricks.

The missing term

When the ? is in the middle

Fill the missing term by confirming the pattern from its neighbours.

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The wrong term

The term that breaks the rule

Spot which term does not fit the pattern of the series.

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5

Choosing the method, and the traps

Clues · traps

A table from what you see to the pattern, and the commonest mistakes.

From the clue to the pattern

What you see tells you

Run every series through this table.

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The commonest mistakes

Five traps

Marks in series are usually lost to hurry.

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6

The practice ladder

Basic → standard → advanced

New questions on every kind of series, in three levels.

Level 1 — Basic

Direct patterns

Series with one plain rule.

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Level 2 — Standard

Two-layer patterns

Gaps of gaps, ×a+b, square±c.

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Level 3 — Advanced

Mixed and woven patterns

Questions at the hard end of competitive exams.

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7

Revision — pattern cards

Flip each card and test yourself

Read the front, think, then flip.

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