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Number Series

Reasoning · Verbal Reasoning

9 Part1 worked examples3 practice questions

What is the next number after 2, 5, 8, 11, 14, …? You say 17 at once, because 3 is added every time. This is a number series: a row of numbers made by a rule, and your job is to catch that rule. Once you catch it, the next term, a missing term in the middle or a wrong term all follow.

In the exam the rule is not announced, so you need a firm way to catch it. The method of this chapter is very simple: always first work out the gap between neighbouring terms (and the ratio), then see how that itself changes. That opens all the rules — equal gaps, growing gaps, multiplying, squares, cubes, primes, alternating rules.

Form of the questionWhat is askedPart
The next term3, 7, 13, 21, 31, ?all parts
A missing term2, 6, ?, 54, 162all parts
The wrong term5, 10, 17, 26, 33, 50 (one term wrong)the wrong-term part

How to use this chapter

  1. Read — each part first explains one rule and then tells how to recognise it.
  2. Study the solved example — in the boxes marked “Example” the full solution is open, step by step.
  3. Try it yourself — in the “Try it yourself first” questions the solution is hidden. Work out the gaps on paper first, then choose.
  4. Know the level — every question is marked Basic, Standard or Advanced.
NoteThe questions you have solved are remembered in this browser. Look at the panel on the right (the “Contents” button on a phone) to see how many you have got right.
1

Foundation — series, terms and rules

Take the gaps → watch the gaps → test the rule

The vocabulary of a series, a four-step method to find the rule and the lists that make recognition quick.

Series, and the method of finding the rule

Four steps that work on every series

The vocabulary of terms, gaps and ratios, four steps to find a rule and the eight families of rules.

Every number of a series is a term. The gap between two neighbouring terms = next term − previous term. The ratio = next term ÷ previous term. In 3, 7, 11, 15 the gaps are 4, 4, 4; in 3, 6, 12, 24 the ratios are 2, 2, 2.

Four steps to find the rule
1. Work out the gaps of neighbouring terms2. Gaps equal? the rule is found; if not look at the gaps of the gaps or the ratio3. Test the rule on all the terms4. Apply the rule at the next step
FamilyHow it showsExample
Equal gapsthe same gap every time4, 9, 14, 19, 24
Changing gapsthe gaps themselves follow a rule3, 4, 6, 9, 13 (+1, +2, +3, +4)
Multiplying / dividingthe same ratio2, 6, 18, 54
Multiply and addthe ratio is almost the same2, 5, 11, 23 (×2 + 1)
Squares, cubesterms near familiar squares or cubes5, 10, 17, 26 (n² + 1)
Primes and known liststerms or gaps are primes, powers of 2, Fibonacci2, 3, 5, 7, 11
Alternatingterms go up and down, or two series are woven5, 20, 8, 17, 11, 14
Sum ruleseach term = the sum of the previous two2, 3, 5, 8, 13
CarefulDo not stop at one gap. Even if the first gap fits, a later term may follow another rule. Always test the rule on at least four or five terms, and after finding the next term run the same rule back over the earlier terms.
Example 1 · Basic
Find the next term of the series: 32, 39, 46, 53, 60, ?
Solution, step by step
  1. 1Write the relation between neighbouring terms: 32 →(+7) 39 →(+7) 46 →(+7) 53 →(+7) 60 →(+7).
  2. 2Rule: the same +7 is added every time (equal gaps).
  3. 3The same rule at the next step: 60 → 67.
Answer 67
If the gaps come out equal, the rule is found.
BasicTry it yourself first
Find the next term of the series: 29, 37, 45, 53, 61, ?
BasicTry it yourself first
Which number fills the gap in this series? 58, 52, ?, 40, 34, 28

The lists that make recognition quick

Squares, cubes, primes, powers of 2, Fibonacci

The lists of numbers most useful in series questions and aids to remember them.

When a term looks like 49, 64, 125 or 343, do not be surprised — they are squares and cubes. The lists below will serve you throughout the chapter. Read them carefully once and now and then write them without looking to test yourself.

Number nn²n³
111
248
3927
41664
525125
636216
749343
864512
981729
101001000
111211331
121441728
131692197
141962744
152253375
ListTerms
Prime numbers2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 …
Powers of 21, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Powers of 31, 3, 9, 27, 81, 243, 729, 2187
Factorials (n!)1, 2, 6, 24, 120, 720, 5040
Fibonacci (each term = the sum of the previous two)1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Triangular (1, 1+2, 1+2+3 …)1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78
Squares of primes4, 9, 25, 49, 121, 169, 289, 361
TrickIf a term is 1 or 2 above or below a square or a cube (like 50 = 49 + 1, 63 = 64 − 1, 126 = 125 + 1), the rule is almost surely “square / cube ± a small number”.
BasicTry it yourself first
Which number fills the gap in this series? 22, 26, ?, 34, 38, 42
2

Adding and subtracting rules

Equal gaps, changing gaps, special gaps

Series in which a number is added or subtracted at every step — and the number is fixed, changes by a rule, or is a square, a cube or a prime.

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Equal gaps

The same number added or subtracted each time

A series with the same gap, recognising it and finding the next or missing term in rising and falling series.

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Changing gaps — the gap of the gaps

+1, +2, +3 … or consecutive odd / even numbers

When the gaps are not equal but themselves grow or shrink by a fixed amount; the adding of consecutive odd and even numbers.

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When the numbers added are special

Primes, squares, cubes and doubling gaps

When the gaps are consecutive primes, squares or cubes of natural numbers, or double or treble every time.

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3

Multiplying and dividing rules

×k, ÷k, ×k ± c and growing multipliers

Series whose terms rise or fall fast because every step multiplies or divides, sometimes together with an addition or subtraction.

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The same multiplier (or divisor) each time

2, 6, 18, 54 … and 1296, 648, 216 …

A geometric series: working out the ratio, finding the next or missing term in rising and falling series.

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Multiply, then add or subtract

Each term = previous × k ± c

When the ratio is nearly but not exactly equal: finding the multiplier k and the fixed number c from the first two steps.

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A growing multiplier and a growing addition

×1 + 1, ×2 + 2, ×3 + 3 … and ×k + 1, ×k + 2 …

When the multiplier or the added number grows by 1 at every step.

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4

Squares, cubes, primes and powers

When the terms themselves lie near a known list

Series whose terms are directly squares, cubes, primes or powers of 2 (sometimes with ±1, ±2).

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Squares, cubes and number × next number

n², n² ± c, n³ ± c, n(n + 1)

Series of consecutive squares and cubes with a small number added or subtracted, and series of products of two consecutive numbers.

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Prime numbers and powers of 2

2, 3, 5, 7 … ; prime ± c ; squares of primes ; 2ⁿ ± c

When the terms are consecutive primes (sometimes ±1, ±2), squares of primes, or consecutive powers of 2 (sometimes ±1, ±2).

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5

Alternating and sum rules

Two series woven together, two operations in turn, the sum of the previous two

Series that do not run in one straight line: the terms go up and down, or every term depends on the previous two.

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Two series woven into one

1st, 3rd, 5th … one series and 2nd, 4th, 6th … another

Splitting the terms into two groups by place and finding a separate rule for each group.

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Two operations in turn

+4, ×2, +4, ×2 … or ×3, −2, ×3, −2 …

When one operation applies at one step and another at the next and the pattern repeats.

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The sum of the previous two terms

Fibonacci and its forms

When every term is made from the sum of the previous two (sometimes ± a small number).

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6

Finding the wrong term

One term is outside the rule — which?

In a series of seven or eight terms one term is wrong; finding the rule from the other terms and catching that term.

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How to catch the wrong term

The term that “jars” in the row of gaps or ratios

Finding the disturbance in the row of gaps, working out the right number in place of the wrong term and confirming it.

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7

Strategy guide

The order of checks, common traps and saving time

The order of checks that works on any series, a table of recognition and the slips to avoid.

The order of checks

What to think on seeing what

A table for guessing the rule from the way the terms move.

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Common traps and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in series questions and how to avoid them.

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8

Mixed practice

All rules mixed — first spot the rule, then solve

The next term, the missing term and the wrong term, in three levels.

Basic practice

One rule, one step

Equal gaps, changing gaps, plain multiplying and simple square series.

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Standard practice

Special gaps, multiply-and-add, primes, alternating

Prime and square gaps, ×k ± c, term lists, two series, two operations and Fibonacci.

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Advanced practice

Cube gaps, growing multipliers, two series, long wrong-term series

Hard series of seven terms and wrong terms among eight, where several possible rules show up together.

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9

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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