Series (Number & Letter)

high importance~4 Q in Tier 18 formulas⚡ 12 shortcuts6 subtopics

Overview Learn (6)Formulas (8)Examples (4)Practice (95)

Number series

⏱ 6 min read🧩 8 question types🎯 26 practice Q

The idea in one minute

Most number-series questions come from eight families. Learn the families, and each question becomes a quick check against a short list. Slow growth points to gaps. Fast growth points to multiplying. Going up and down points to two series mixed together.

The eight families

Almost every number series uses one of these rules.

Family	Example	Next
Same or growing gap	7, 12, 17, 22	27
Multiply, then adjust	3, 7, 15, 31	63
Squares and cubes	3, 8, 15, 24	35
Gaps of gaps	2, 5, 11, 21, 36	57
Famous gaps	10, 11, 13, 17, 25	41
Alternate series	3, 50, 6, 45, 12, 40	24
Mixed steps	2, 3, 8, 27, 112	565
Sum of earlier terms	2, 3, 5, 8, 13	21

Add the same or a growing number

In 7, 12, 17, 22 the gap is always 5. This is an arithmetic progression (AP).

The gap can also grow. In 11, 13, 17, 23, 31 the gaps are 2, 4, 6, 8. The next gap is 10, so the next term is 41. The gap can fall too: 120, 119, 116, 111 has gaps 1, 3, 5, all taken away.

Multiply, then add or subtract

Multiply by a fixed number, then add or subtract a small number. The series 3, 6, 12, 24 is ×2. The series 3, 7, 15, 31 is ×2 + 1. The series 4, 10, 28, 82 is ×3 − 2.

Tip: Divide a big term by the one before it. If the answer is near 2 or 3, try ×2 or ×3. The small amount left over is the number to add or subtract.

Squares and cubes

Learn squares up to 30 and cubes up to 15. Look for terms just above or below them.

3, 8, 15, 24, 35 is $n^2 - 1$ .
2, 9, 28, 65 is $n^3 + 1$ .
2, 6, 12, 20, 30 is $n \times (n + 1)$ .

Gaps of gaps, and famous gaps

Sometimes the first gaps are not equal, but the second gaps follow a rule.

Example: 2, 5, 11, 21, 36. First gaps: 3, 6, 10, 15. Second gaps: 3, 4, 5. The next second gap is 6, so the next first gap is 21. The next term is 36 + 21 = 57.

The gaps can also be squares (1, 4, 9, 16), cubes (1, 8, 27, 64), primes or powers of 2. In 10, 11, 13, 17, 25 the gaps are 1, 2, 4, 8. The next gap is 16, so the next term is 41.

Alternate series

Two series are mixed. Places 1, 3, 5, 7 make one series. Places 2, 4, 6, 8 make another.

In 3, 50, 6, 45, 12, 40, 24, ? the odd places are 3, 6, 12, 24 (×2). The even places are 50, 45, 40 (−5). The eighth term is an even place, so it is 35.

Watch: Suspect an alternate series when the terms jump up and down.

Mixed steps and sums of earlier terms

The step changes in a fixed way.

×1 + 1, ×2 + 2, ×3 + 3: 2, 3, 8, 27, 112. The next term is 112 × 5 + 5 = 565.
Two steps take turns: 5, 6, 12, 13, 26, 27. The steps are +1, ×2, +1, ×2, so the next term is 54.
Signs change: 10, 12, 9, 13, 8. The steps are +2, −3, +4, −5, so the next step is +6 and the term is 14.

In a sum-of-earlier-terms series, each term is the sum of the last two: 2, 3, 5, 8, 13, 21. Or of the last three: 1, 2, 3, 6, 11, 20, 37.

How to choose fast

Slow growth: use the ladder of gaps.
Fast growth: divide each term by the last one.
Up and down: split into odd and even places.
Terms near squares or cubes: use the family of squares and cubes.
Each term close to the sum of the two before: sum of earlier terms.

Question types you will see

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

Type 1very common2 practice Q

Same or growing gap

How to spot it:

The terms grow slowly and steadily. The gaps are equal, or they rise or fall by the same amount.

Method

Write the gaps under the series.
If the gaps are equal, add one more gap.
If the gaps rise evenly, work out the next gap first.

Why it works:

A steady gap and an evenly growing gap are the two simplest families.

Try this

Find the next term: 4, 6, 10, 16, 24, ?

Show solution

Gaps: 2, 4, 6, 8.
The next gap is 10.
24 + 10 = 34.

Answer

Type 2very common2 practice Q

Multiply, then add or subtract

How to spot it:

The terms roughly double or triple. The ratio is close to, but not exactly, the same.

a_{n+1} = k \cdot a_n \pm r

Method

Divide a big term by the one before it.
Try ×2 or ×3 on a small term.
Find the number left over. That is r.
Check the rule on every term.

Why it works:

Multiplying with a small fix at each step grows fast and looks messy.

Try this

Find the next term: 2, 5, 14, 41, ?

Show solution

Try ×3: 2 × 3 = 6, and 5 is 1 less.
14 = 5 × 3 − 1 and 41 = 14 × 3 − 1.
Next: 41 × 3 − 1 = 122.

Answer

122

Type 3very common3 practice Q

Squares, cubes and n × (n + 1)

How to spot it:

The terms sit on or next to squares (4, 9, 16), cubes (8, 27, 64), or look like 2, 6, 12, 20.

n^2 \pm k,\ n^3 \pm k,\ n(n+1)

Method

Compare each term with the nearest square or cube.
Find the small number added or taken away.
Check that it is the same for every term.
Use the next n.

Why it works:

Setters like squares and cubes because the terms grow in a way students can check.

Try this

Find the next term: 0, 3, 8, 15, 24, ?

Show solution

$1^2 - 1 = 0$ , $2^2 - 1 = 3$ , $3^2 - 1 = 8$ .
$4^2 - 1 = 15$ and $5^2 - 1 = 24$ .
Next: $6^2 - 1 = 35$ .

Answer

Type 4common2 practice Q

Gaps of gaps

How to spot it:

The first gaps are not equal, but they rise in a pattern like 1, 3, 6, 10.

Method

Write the first gaps.
Write the second gaps.
Continue the second gaps by one step.
Climb back to the next gap, then the next term.

Why it works:

If the second gaps follow a rule, the first gaps are easy to extend.

Try this

Find the next term: 3, 4, 7, 13, 23, ?

Show solution

First gaps: 1, 3, 6, 10.
Second gaps: 2, 3, 4. The next is 5.
Next first gap = 10 + 5 = 15. Term = 23 + 15 = 38.

Answer

Type 5common2 practice Q

Two series mixed together

How to spot it:

The series goes up and down, or big and small terms take turns.

Method

Write odd places on one line and even places on another.
Solve each line on its own.
Find whether the asked place is odd or even.
Extend that line by one term.

Why it works:

Two simple series woven together look like one messy series.

Try this

Find the next term: 1, 10, 2, 20, 3, 30, ?

Show solution

Odd places: 1, 2, 3. Even places: 10, 20, 30.
The seventh term is an odd place.
The odd line goes up by 1, so the term is 4.

Answer

Type 6common3 practice Q

Mixed steps: ×1 + 1, ×2 + 2 and so on

How to spot it:

The terms grow faster and faster with no fixed ratio, or two steps clearly take turns.

Method

Try ×1 + 1 on the first pair.
If the multiplier and the added number both rise by 1 each step, this is the rule.
Check every pair, then apply the next multiplier.

Why it works:

The multiplier keeps growing, so the ratio is never the same twice.

Try this

Find the next term: 1, 2, 6, 21, 88, ?

Show solution

1 × 1 + 1 = 2 and 2 × 2 + 2 = 6.
6 × 3 + 3 = 21 and 21 × 4 + 4 = 88.
Next: 88 × 5 + 5 = 445.

Answer

445

Type 7occasional2 practice Q

Sum of the earlier terms

How to spot it:

Each term is close to the sum of the two (or three) terms before it.

Method

Add the first two terms and compare with the third.
If it fits, add the last two terms for the next.
If the terms are small changes off, try the last three.

Why it works:

In a Fibonacci-type series, the gaps look just like the series itself.

Try this

Find the next term: 1, 1, 2, 3, 5, 8, ?

Show solution

1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8.
Each term is the sum of the last two.
5 + 8 = 13.

Answer

Type 8occasional2 practice Q

Gaps that are squares, cubes, primes or powers of 2

How to spot it:

The gap row is a famous list: 1, 4, 9, 16, or 1, 8, 27, or 2, 3, 5, 7, or 1, 2, 4, 8.

Method

Write the first gaps.
Recognise the list of the gaps.
Take the next number of that list.
Add it to the last term.

Why it works:

The gaps carry the rule, not the terms.

Try this

Find the next term: 5, 6, 10, 19, 35, ?

Show solution

Gaps: 1, 4, 9, 16. These are squares.
The next square is 25.
35 + 25 = 60.

Answer

Formula sheet

Same gap (AP)

a_n = a_1 + (n - 1)d

d is the equal gap from the ladder.

Multiply then adjust

a_{n+1} = k \cdot a_n \pm r

Try k = 2 and 3 first, with r = 1 or 2.

Squares family

a_n = n^2 \pm k \ \text{ or } \ n^2 + n

If the second gaps are all 2, the series is built on n squared.

Same ratio (GP)

a_n = a_1 \cdot r^{\,n-1}

r is the fixed ratio. Test ×2, ×3 and ×1.5.

Shortcuts that save time

⚡ Ratio test before anything fancy

Divide each term by the one before it. If the result is close to 2 or 3, try ×k ± r. Find r from what is left over.

Example

Find the next term: 5, 11, 23, 47, ?

Show solution

11 ÷ 5, 23 ÷ 11 and 47 ÷ 23 are all near 2.
5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47.
47 × 2 + 1 = 95.

Answer

⚡ Split a jumpy series

If the terms go up and down, write the odd places on one line and the even places on another. Solve each line on its own. Then find which line the asked place belongs to.

Example

Find the seventh term: 12, 5, 14, 8, 16, 11, ?

Show solution

Odd places: 12, 14, 16.
Even places: 5, 8, 11.
The seventh term is an odd place, so it is 16 + 2 = 18.

Answer

⚡ Cubes hiding in the gaps

If the gaps grow very fast, check for cubes (1, 8, 27, 64, 125) or powers of 2 (1, 2, 4, 8, 16).

Example

Find the next term: 2, 3, 11, 38, 102, ?

Show solution

Gaps: 1, 8, 27, 64. These are cubes.
The next gap is 5³ = 125.
102 + 125 = 227.

Answer

227

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Trying only 'add' on a series that grows fast.

Look at the size first. Fast growth needs a ratio test.

Mistake 02

Stopping at the first row of gaps.

Take the gaps of the gaps. Look for squares, cubes and powers of 2.

Mistake 03

Reading an up-and-down series as one series.

Split into odd and even places. Then find the place asked.

Mistake 04

Using the last two terms to guess the rule.

Test the rule on every term from the start.

Mistake 05

Forgetting that ×k ± r needs a small leftover.

After ×2 or ×3, find what is left over. That is the r.

Mistake 06

Giving the second-gap answer as the next term.

Climb back up: next second gap, then next gap, then next term.

Quick revision

Read this the night before the exam.

Slow growth: gaps. Fast growth: ratio. Up and down: two series. Near squares or cubes: powers.
Same or growing gap: 7, 12, 17 (+5) and 11, 13, 17, 23 (+2, +4, +6).
Multiply then adjust: 3, 7, 15, 31 is ×2 + 1.
Squares and cubes: n² − 1, n³ + 1 and n(n + 1).
Gaps of gaps: find the next second gap, then the next gap, then the term.
Alternate series: split odd and even places and check which one the asked term is in.
Sum of earlier terms: each term is the sum of the last two, or three, terms.

Practice: 26 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.

Continue learning