Sequences & Progressions

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Pattern Sequences: Next, Missing & Wrong Terms

⏱ 4 min read🧩 6 question types🎯 14 practice Q

The idea in one minute

A pattern sequence hides a rule, and the question asks for the next term, a missing term, or the wrong term. The work is a fixed routine: take differences, check ratios, split alternating chains, and test position rules like $n^2$ or $2^n$ .

What a pattern sequence asks

A list of numbers hides a rule. The question asks for the next term, a missing middle term, or the one wrong term.

There is no formula to memorise. There is a routine: test the common rules in order until one fits every term.

Rule: A correct rule must match every given term. Two matching terms prove nothing.

The difference ladder

Subtract each term from the next. If the differences are equal, it is an AP — add the difference once more.

If the differences differ, take their differences too (the second differences).

2, 5, 10, 17, 26: the gaps are 3, 5, 7, 9 — the odd numbers. Next gap is 11, so the next term is $26 + 11 = 37$ .

The same list is $n^2 + 1$ : $1^2+1$ , $2^2+1$ , $3^2+1$ , ... Both roads give 37.

3, 4, 8, 17, 33: gaps are 1, 4, 9, 16 — the squares. Next gap 25 gives 58.

Tip: Odd-number gaps, square gaps and doubling gaps cover most exam series.

Multiply and mixed operations

When the list grows fast, try division instead of subtraction.

5, 10, 20, 40: each term is double the one before, so the next is 80.

2, 4, 12, 48, 240: the multipliers are $\times 2$ , $\times 3$ , $\times 4$ , $\times 5$ , so the next multiplier is 6 and the next term is 1440.

Mixed rules multiply and add: 5, 11, 23, 47 uses " $\times 2 + 1$ " each time, giving 95 next.

Watch: A ratio check (divide neighbours) catches multiplication; a difference check cannot.

Two alternating chains

Odd-position terms and even-position terms can follow different rules.

5, 9, 7, 11, 9, 13: positions 1, 3, 5 give 5, 7, 9 ( $+2$ ). Positions 2, 4, 6 give 9, 11, 13 ( $+2$ ). The 7th term belongs to the first chain: 11.

7, 12, 10, 15, 13, 18: chains $7, 10, 13$ and $12, 15, 18$ , both $+3$ . Next is 16.

Tip: If one single rule fails after two or three terms, split odd and even positions before anything else.

Position-based rules

Many series tie each term to its position $n$ :

$n^2$ : 1, 4, 9, 16.
$n^3$ : 1, 8, 27, 64.
$n^2 - 1$ : 3, 8, 15, 24, 35.
$n^n$ : 1, 4, 27, 256.

3, 8, 15, 24, 35: these are $2^2-1$ , $3^2-1$ , $4^2-1$ , ..., so the next is $7^2 - 1 = 48$ .

Prime numbers 2, 3, 5, 7, 11 and their $\pm 1$ shifts appear often too.

Missing and wrong terms

A missing middle term uses the same routine — the rule must fit both neighbours.

7, 14, 28, ?, 112: each term doubles, so the missing term is 56.

2, 3, 6, ?, 14, 15, 30: the rule alternates $+1$ and $\times 2$ , so the missing term is 7.

For a wrong-number question, build the correct series from the rule, then point at the term that breaks it.

2, 6, 12, 20, 30, 40, 56: the rule is $n(n+1)$ , so the sixth term should be 42. The wrong number is 40.

Example: 5, 11, 23, 47, 94, 191 — the rule " $\times 2 + 1$ " gives 95 in fifth place, so 94 is wrong.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Next term by the difference ladder

How to spot it:

A plain list of growing numbers; the gaps themselves show a pattern.

Method

Write the first differences.
If needed, write the second differences.
Extend the gap pattern and add.

Why it works:

Differences strip one layer of the rule, and the layer beneath is usually a familiar list.

Try this

Find the next term: 2, 5, 10, 17, 26, ...

Show solution

Gaps: 3, 5, 7, 9.
Next gap 11 gives $26 + 11$ .

Answer

Type 2common2 practice Q

Next term by multiply or divide

How to spot it:

The list grows (or shrinks) fast; dividing neighbours gives equal or stepping ratios.

Method

Divide each term by the one before.
Read the ratio pattern (equal, or 2, 3, 4, ...).
Apply the next ratio.

Why it works:

Constant or stepping multipliers show up only under division, never under subtraction.

Try this

Find the next term: 2, 4, 12, 48, 240, ...

Show solution

Ratios: $\times 2$ , $\times 3$ , $\times 4$ , $\times 5$ .
Next: $240 \times 6$ .

Answer

1440

Type 3common2 practice Q

Two alternating chains

How to spot it:

The numbers jump up and down, or one clean rule fits only every other term.

Method

Split into odd and even positions.
Find each chain's rule.
Answer from the chain that owns the asked position.

Why it works:

Two interleaved simple rules look like chaos until the positions are separated.

Try this

Find the next term: 5, 9, 7, 11, 9, 13, ...

Show solution

Odd positions: 5, 7, 9; even: 9, 11, 13.
7th term: odd chain $+2$ gives 11.

Answer

Type 4common3 practice Q

Squares, cubes and position rules

How to spot it:

The terms sit near familiar squares or cubes — one less, one more, or the cubes themselves.

Method

Compare each term with $n^2$ and $n^3$ .
Find the fixed offset ( $\pm 1$ , $\pm 2$ ...).
Apply the rule at the next position.

Why it works:

Offset squares and cubes are the most reused hidden rules in exam series.

Try this

Find the next term: 3, 8, 15, 24, 35, ...

Show solution

$2^2-1$ , $3^2-1$ , $4^2-1$ , ...
Next: $7^2 - 1 = 48$ .

Answer

Type 5very common2 practice Q

Missing term inside the series

How to spot it:

A '?' sits between two given terms instead of at the end.

Method

Find a rule that fits every known term.
Apply it at the missing position.
Check the answer against both neighbours.

Why it works:

A middle term must satisfy the rule from both sides, which makes it a strong check.

Try this

Find the missing term: 7, 14, 28, ?, 112

Show solution

Each term doubles.
Missing term: $28 \times 2 = 56$ .

Answer

Type 6common2 practice Q

Wrong number in the series

How to spot it:

'Find the wrong number' — one listed term breaks the rule the others follow.

Method

Find the rule from the terms that agree.
Build the correct series.
Point at the position that breaks it.

Why it works:

The rule fixed by the majority of terms exposes the single misfit.

Try this

Find the wrong number: 2, 6, 12, 20, 30, 40, 56

Show solution

Rule $n(n+1)$ : 2, 6, 12, 20, 30, 42, 56.
The sixth term is 40 — wrong.

Answer

Formula sheet

Common position rules

n^2,\ n^3,\ n^2+1,\ n^2-1,\ 2^n,\ n(n+1)

Compare each term with its position $n$.

Multiply-and-add rule

a_{n+1} = a_n \times r + c

A frequent hidden rule; test it when differences fail.

Shortcuts that save time

⚡ Take differences, then differences again

The first differences often hide a second pattern — odd numbers, squares, doubling. Two ladders solve most series.

Example

Find the next term: 2, 5, 10, 17, 26, ...

Show solution

Gaps: 3, 5, 7, 9 (odd numbers).
Next gap 11: $26 + 11 = 37$ .

Answer

⚡ Split alternating series into two chains

One rule for odd positions, one for even positions. Answer from the chain the asked position belongs to.

Example

Find the next term: 5, 9, 7, 11, 9, 13, ...

Show solution

Odd positions: 5, 7, 9 ( $+2$ ).
7th term continues it: 11.

Answer

⚡ Test the position rules

Compare each term with $n^2$ , $n^3$ , $n^2 \pm 1$ , $2^n$ and primes at its position. One of them usually snaps into place.

Example

Find the next term: 3, 8, 15, 24, 35, ...

Show solution

Terms are $2^2-1$ , $3^2-1$ , $4^2-1$ , ...
Next: $7^2 - 1 = 48$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Testing only addition — many series multiply, alternate two operations, or follow squares and cubes.

Mistake 02

Stopping at the first difference — the differences themselves often form a pattern (second differences).

Mistake 03

Applying one chain's rule to the other in an alternating series — split the series first.

Mistake 04

Ignoring the position of each term — rules like $n^2 + 1$ tie a term to its place in the list.

Mistake 05

Changing two things at once while testing a guessed rule — a valid rule must fit every given term unchanged.

Quick revision

Read this the night before the exam.

Routine: differences, second differences, ratios, alternating chains, position rules.
Odd gaps, square gaps, doubling gaps cover most series.
Fast growth: divide neighbours to find the multiplier.
Jumping lists: split odd and even positions.
A rule must fit every term; one misfit marks the wrong number.
Common position rules: $n^2$ , $n^3$ , $n^2 \pm 1$ , $2^n$ , $n(n+1)$ , primes.

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.

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