Odd Man Out and Series: Formulas, Shortcuts and Solved Examples

What is a number series?

A series is a list of numbers that follow a hidden rule.

You are asked to find the next term, or the term that breaks the rule.

Your job is to find the rule first. Then apply it.

What are the main kinds of series?

Arithmetic: the difference between neighbours stays the same.
Geometric: the ratio between neighbours stays the same.
Squares and cubes: the terms are consecutive perfect squares or cubes.
Alternating: two rules take turns, such as add 8 then subtract 4.

What does “odd man out” mean?

Four items share a property and one does not. You find the one that breaks the pattern.

In 8, 27, 64, 100, 125, all but 100 are perfect cubes. So 100 is the odd one out.

In 3, 5, 7, 9, 11, all but 9 are primes. So 9 is the odd one out.

What if the gaps keep changing?

Look at the gaps themselves. Take the series 1, 2, 4, 7, 11.

The gaps are 1, 2, 3, 4. They grow by 1 each time.

So the next gap is 5, and the next term is 16.

This is a “second-level” pattern. It is common in harder questions.

What is a mixed or alternating series?

Some series hide two patterns, woven together. Read the odd places and the even places as two lists.

Take 3, 10, 6, 20, 12, 40. The odd places give 3, 6, 12. They double each time.

The even places give 10, 20, 40. They also double.

So the next terms are 24 and 80.

How do I find the rule?

Subtract neighbours first. If the gaps match, it is arithmetic.If the gaps do not match, divide neighbours. If the ratios match, it is geometric.

Why it works: these are the two simplest patterns, and they cover most exam series.

Which squares and cubes must I know?

Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169.
Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331.

Why it helps: if the terms look like these, you only need the next one on the list.

How do I find the wrong number?

Find the rule from the terms that agree. Then check every term.

The wrong term is the one the rule does not give.

Why it works: only one term is wrong. So the other terms still follow the rule.

How do I read an alternating series?

Look at the gaps. They repeat in a pattern, such as +8, −4, +8, −4.

The term that breaks the repeating gap is the wrong one.

Which other patterns appear?

Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23.
Factorials: 1, 2, 6, 24, 120.
Squares plus or minus one: 3, 8, 15, 24 come from n² − 1.

Why it helps: these lists are short. So you can spot them by eye.

How do I find a far term quickly?

For an arithmetic series, the nth term is first term + (n − 1) × gap.

For 7, 16, 25, the tenth term is 7 + 9 × 9 = 88.

Why it works: you add the gap once for every step after the first term.

Easy, step by step: find the next number in 7, 16, 25, 34, 43, 52.

Step 1: subtract neighbours. 16 − 7 = 9, 25 − 16 = 9.

Step 2: every gap is 9, so the series is arithmetic.

Step 3: add 9 to the last term. 52 + 9 = 61.

The answer is 61.

Easy, step by step: find the next number in 2, 8, 32, 128, 512.

Step 1: the gaps are 6, 24 and 96. They do not match.

Step 2: divide neighbours. 8 ÷ 2 = 4 and 32 ÷ 8 = 4.

Step 3: multiply the last term by 4. 512 × 4 = 2048.

The answer is 2048.

Medium, step by step: find the next number in 216, 343, 512, 729.

Step 1: 216 is 6³, 343 is 7³, 512 is 8³ and 729 is 9³.

Step 2: the next base is 10.

Step 3: 10³ = 1000.

The answer is 1000.

Medium, step by step: find the wrong number in 6, 14, 22, 32, 38, 46, 54.

Step 1: the gaps are 8, 8, 10, 6, 8, 8.

Step 2: the rule is add 8. From 22 the next term should be 30.

Step 3: the term after 30 is 38, which fits. So 32 is the wrong one.

The answer is 32.

Exam level, step by step: find the wrong number in 4, 12, 36, 180, 324, 972.

Step 1: 12 ÷ 4 = 3 and 36 ÷ 12 = 3.

Step 2: the rule is multiply by 3. After 36 comes 108.

Step 3: 108 × 3 = 324 and 324 × 3 = 972. Both fit.

The wrong number is 180.

Exam level, step by step: find the next number in 3, 10, 6, 20, 12, 40.

Step 1: the gaps jump up and down, so no single gap rule fits.

Step 2: split the places. Odd places give 3, 6, 12. Even places give 10, 20, 40.

Step 3: both lists double. The next odd-place term is 24.

The answer is 24.

Exam level, step by step: find the odd one out in 8, 27, 64, 100, 125.

Step 1: test for cubes. 8 is 2³, 27 is 3³ and 64 is 4³.

Step 2: 125 is 5³. But 100 is not a cube, since 4³ is 64 and 5³ is 125.

The odd one out is 100.

Which traps cost marks?

Checking only the first gap and assuming the rule.
Mistaking a squares series for an arithmetic one.
Reading a wrong term as the new rule.
Forgetting that a series can have two rules taking turns.
Making an arithmetic slip while multiplying.

Which shortcuts save time?

Check gaps first, then ratios, then squares and cubes.
If the gaps themselves grow by a fixed amount, think of squares.
For the wrong number, test the terms around the odd gap.
Memorise the squares and cubes up to 15 and 10.

What patterns come up in exams?

Five patterns repeat. Find the next term, spot the wrong number, spot the odd man out, name the series type and read an alternating series.

Each takes about a minute if you test gaps and ratios first.

How do I avoid the gap trap?

A wrong term spoils two gaps, not one. So do not blame the wrong gap.

Find the rule from clean gaps far apart. Then test each term against it.