IMAT Logical Reasoning · Spatial and Sequence Reasoning

Number Sequences: Finding the Rule

A number sequence follows a rule; you find it by testing, in order, a constant difference, a constant ratio, a pattern in the differences, and a rule built from earlier terms.

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Why this matters

The 2026 ministry paper asked for a missing term in a sequence where each term is built from the two terms before it. It is one of the few items in this chapter that needs no figure at all.

Concept 1 of 3: Arithmetic and geometric sequences

The two simplest rules are "add the same number each time" and "multiply by the same number each time". Check the gaps between terms: equal gaps mean adding, equal ratios mean multiplying. Once you know the rule, a formula jumps straight to any term without listing them all.

Definition

An arithmetic sequence adds a fixed common difference $d$ : for example 7, 11, 15, 19 has $d = 4$ .
A geometric sequence multiplies by a fixed common ratio $r$ : for example 3, 6, 12, 24 has $r = 2$ .
To reach the $n$ th term you take $n - 1$ steps from the first term.
A ratio can be a fraction (81, 27, 9 has $r = \tfrac{1}{3}$ ) or negative (2, $-6$ , 18 has $r = -3$ ).

The nth term

a_n = a_1 + (n - 1)\,d \qquad a_n = a_1\, r^{\,n - 1}

$a_1$ first term
$d$ common difference (arithmetic)
$r$ common ratio (geometric)
$n$ position of the term you want

Worked example

Find the 20th term of 7, 11, 15, 19, … and the 8th term of 3, 6, 12, 24, …

Practice this conceptself-check · 3 quick reps

The nth term is n minus 1 steps away

The first term is already in place, so the 7th term is only 6 steps on. Using

r^n

n \times d

gives the next term after the one asked for, and that value is usually among the options.

Concept 2 of 3: Differences of differences, and interleaved sequences

When the gaps are not equal, write the gaps down as a new sequence. Very often that new row is simple: equal, growing by a fixed amount, or doubling. Another common trick is two sequences shuffled together, one in the odd positions and one in the even positions.

Definition

The first differences are the gaps between neighbouring terms. If they are constant, the sequence is arithmetic.
If the first differences themselves change by a constant, the second differences are constant (squares 1, 4, 9, 16 behave like this).
The differences may also form a geometric sequence (gaps 1, 2, 4, 8).
An interleaved (alternating) sequence hides two rules: read terms 1, 3, 5, … and terms 2, 4, 6, … separately.

Worked example

Find the next term of 3, 4, 7, 12, 19, … and the next two terms of 2, 20, 4, 17, 8, 14, …

Practice this conceptself-check · 3 quick reps

Check the rule on every given term

A rule that fits the first three terms can fail at the fourth. Before choosing an option, run your rule through all the terms shown; the wrong options are built from rules that fit only the beginning.

Concept 3 of 3: Recurrence rules: each term built from the terms before it

Some sequences grow in a way no single difference or ratio explains. Then each term is usually made from the one or two terms before it: the sum of the last two (Fibonacci), or twice the last plus the one before. Test a guess on three or four terms in a row.

Definition

A recurrence gives each term from earlier terms.

Fibonacci-like: $a_n = a_{n-1} + a_{n-2}$ , for example 1, 3, 4, 7, 11, 18.
Weighted versions such as $a_n = 2a_{n-1} + a_{n-2}$ or $a_n = a_{n-1} + 2a_{n-2}$ .
One-term rules such as $a_n = 2a_{n-1} + 1$ (1, 3, 7, 15, 31) or "multiply by 2, then 3, then 4" (1, 2, 6, 24).
Signs: the differences do not settle, and the ratio of neighbouring terms creeps towards a fixed value.
With a gap in the middle, use the terms on BOTH sides of it to check your value.

Two-term recurrence

a_n = p\,a_{n-1} + q\,a_{n-2}

$p, q$ fixed whole numbers, often 1 or 2
$a_{n-1}, a_{n-2}$ the previous term and the one before it

Worked example

Find the next term of 1, 1, 3, 7, 17, …

Practice this conceptself-check · 3 quick reps

Use the term after the gap

When a missing term sits in the middle, the terms after it are a free check. A value that fits the rule on one side but not the other is the wrong rule, and IMAT distractors are often exactly those values.

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 (2)

Arithmetic and geometric sequences
The nth term
$a_n = a_1 + (n - 1)\,d \qquad a_n = a_1\, r^{\,n - 1}$
Recurrence rules: each term built from the terms before it
Two-term recurrence
$a_n = p\,a_{n-1} + q\,a_{n-2}$

Watch out for (3)

The nth term is n minus 1 steps away
→ Arithmetic and geometric sequences
Check the rule on every given term
→ Differences of differences, and interleaved sequences
Use the term after the gap
→ Recurrence rules: each term built from the terms before it

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