SPI Reasoning · Series

Number Series

A number series follows one rule from term to term. Find the rule from the differences, the multipliers or the squares, then use it to fill the gap or catch the wrong term.

All Series notes

Why this matters

Five questions in the bank, all from the 2022 paper: three EASY, one MODERATE and one HARD. Two ask for a missing term and three for the wrong term, and each yields to one of three checks: the differences, the multipliers, or squares with one added or taken away.

Concept 1 of 3: Write the differences first

Under every series is a second, simpler row: the gaps between the terms. Write it out. Usually the gaps are all the same, or grow by a fixed step, or are a list you know, such as the primes.

Definition

The method:

Write the difference between each pair of next-door terms under the series.
Same each time: add that number (5, 10, 15, 20 adds 5).
Growing by a fixed step: the differences go up by 2 or 3 each time.
A known list: the differences are 2, 3, 5, 7, 11 (primes) or 1, 4, 9, 16 (squares).
For a wrong term, find the one place where the pattern of differences breaks. One wrong term spoils two differences next to it.

Next term from the differences

t_{n+1} = t_n + d_n

tₙthe term in place n
dₙthe difference that follows it

Worked example

Find the next term: 3, 7, 13, 21, 31, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q31Hard

Example 1 · Series · Number Series

Find out the wrong term in the given series in the next three questions.

23, 34, 49, 64, 83, 106, 135, 166.

Blaming the wrong term of two

One wrong term spoils the difference before it and the one after it. Fix the term in the middle of the two odd differences, then check the whole row again.

Not recognising the primes

Differences such as 5, 7, 11, 13 have no fixed step. They are consecutive primes, so the next one is 17, not 15.

Drill 1 more on write the differences first

Concept 2 of 3: When the terms grow fast, divide

If the terms jump a long way, the differences are messy. Divide each term by the one before it instead. The multiplier is often the same each time or goes up by one.

Definition

Divide each term by the term before it.
Same multiplier: 3, 6, 12, 24 doubles each time.
Multiplier rising by one: $\times 2, \times 3, \times 4$ and so on.
For a missing term in the middle, check your answer against the term after the gap too.

Next term from the multiplier

t_{n+1} = t_n \times r_n

rₙthe multiplier from place n to place n + 1

Worked example

Find the next term: 2, 4, 12, 48, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q29Easy

Example 2 · Series · Number Series

Complete the series 6,18,72, ?, 2160.

Not checking the term after the gap

A missing term must lead on to the term after it by the same rule. If your answer does not, the rule is wrong.

Keeping the multiplier fixed

When

\times 3

is followed by

\times 4

, the next step is

\times 5

. Multiplying by 4 again gives a printed wrong option.

Concept 3 of 3: Squares, cubes and two series in one

Some series are squares or cubes with a small number added or taken away. Others are two series woven together: read the odd places and the even places separately and each is simple.

Definition

Know the squares 1, 4, 9, 16, 25, 36, 49 and the cubes 1, 8, 27, 64, 125.
If each term is one more or one less than a square, the rule is $n^2 + 1$ or $n^2 - 1$ .
If the terms go up and down, split them: places 1, 3, 5 make one series and places 2, 4, 6 make another.
Use the other series to check your answer.

Square and cube rules

t_n = n^2 \pm k \qquad t_n = n^3 \pm k

Worked example

Find the next term: 1, 10, 3, 20, 5, 30, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q22Moderate

Example 3 · Series · Number Series

Find the value of x in the series 5, 3, 10, 8, 17, 15,x , 24.

Reading an up-and-down series as one

If the terms rise and fall in turn, the differences make no sense as one row. Split the odd and even places first.

Missing a square by one

A term one away from the rule, such as 50 where 51 fits

n^2 + 2

, is easy to pass over. Write the square next to each term to see it.

Drill 1 more on squares, cubes and two series in one

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Write the differences first
Next term from the differences
$t_{n+1} = t_n + d_n$
When the terms grow fast, divide
Next term from the multiplier
$t_{n+1} = t_n \times r_n$
Squares, cubes and two series in one
Square and cube rules
$t_n = n^2 \pm k \qquad t_n = n^3 \pm k$

Watch out for (6)

Blaming the wrong term of two
→ Write the differences first
Not recognising the primes
→ Write the differences first
Not checking the term after the gap
→ When the terms grow fast, divide
Keeping the multiplier fixed
→ When the terms grow fast, divide
Reading an up-and-down series as one
→ Squares, cubes and two series in one
Missing a square by one
→ Squares, cubes and two series in one

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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