Series (Number & Letter)
🔒 Log in to trackHow every series is cracked
🔒 Log in to trackA series is a list of numbers written in a fixed order. Every term comes from the one before it by one small rule. Your job is to name that rule, then find the next term or the missing term. The difference ladder finds the rule in most questions: subtract neighbours, and subtract again if needed.
What a series is
A series is a list of numbers in a fixed order. Each term comes from the one before it by the same small rule.
Think of a savings box. You save ₹100, ₹150 and ₹200. Next you save ₹250, because the rule is "add ₹50".
Exams ask series in three forms.
- Next term: 4, 9, 16, 25, ? Find the term after the last one.
- Missing term: 4, 9, ?, 25, 36. A term in the middle is hidden.
- Wrong term: one term in a full series is wrong. The wrong number lesson covers it.
Look at the size first
Before you calculate, see how fast the terms grow.
| What you see | Likely rule |
|---|---|
| 5, 9, 13, 17: slow and steady | Add a number |
| 3, 7, 15, 31: nearly doubling | Multiply, maybe add a little |
| 8, 20, 10, 18, 12: up and down | Two series mixed together |
| 3, 8, 15, 24: just below 4, 9, 16, 25 | Squares |
The difference ladder
Write the gap between each pair of neighbours. This is the first row of gaps.
Example: 2, 5, 10, 17, 26. The gaps are 3, 5, 7, 9. They rise by 2, so the next gap is 11. The next term is 26 + 11 = 37.
If the gaps are not clear, take the gaps of the gaps. Most exam series become clear in two rows.
| What the gaps show | Rule |
|---|---|
| All gaps equal | Add the same number |
| Gaps rise or fall evenly | Growing or shrinking gap |
| Gaps double | Powers of 2 are added |
| Gaps are 1, 4, 9, 16 | Squares are added |
When the ladder fails
Try these in order.
- Ratio: divide each term by the one before it.
- Alternate terms: read places 1, 3, 5 and places 2, 4, 6 as two series.
- Squares and cubes: look for terms near them.
- Mixed steps: such as ×1 + 1, ×2 + 2, ×3 + 3.
The missing term in the middle
Use the terms on both sides of the gap to find the rule. Then check twice. Your term must come from the one before it. The next term must come from your term.
Example: 3, 7, ?, 31, 63. Each term is ×2 + 1. So ? = 7 × 2 + 1 = 15. Check: 15 × 2 + 1 = 31.
Fractions, decimals and halving
Do not panic at decimals. The series 64, 32, 16, 8, 4 halves each time, so the next term is 2. The series 0.5, 1.5, 4.5, 13.5 is ×3, so the next term is 40.5.
For fractions, treat the tops and the bottoms as two series. In the tops go 1, 3, 5 and the bottoms go 3, 5, 7. Both rise by 2, so the next term is .
Two missing terms
Find the rule from the visible terms. Fill the first blank, then move on to the second. Check the term after each blank.
Tip: Your rule must fit every term you can see, not just the last two.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Find the next term with the difference ladder
The question shows five to seven terms ending in a '?'. Nothing is hidden in the middle.
Write the gaps under the series.
If the gaps are not equal, write the gaps of the gaps.
Continue the last row by one step.
Climb back up: add to the last gap, then to the last term.
The ladder turns a hard series into a short list of simple gaps.
Find the next term: 6, 11, 21, 36, 56, ?
Show solutionHide solution
Gaps: 5, 10, 15, 20.
The gaps rise by 5, so the next gap is 25.
56 + 25 = 81.
81
A missing term in the middle
A '?' sits between the given terms, like 7, 15, ?, 63, 127.
Use the terms on both sides of the gap to find the rule.
Find the missing term from the term before it.
Check that the next term comes from your answer.
The terms on both sides fix the rule, and the second check catches slips.
Find the missing term: 7, 15, ?, 63, 127
Show solutionHide solution
15 = 7 × 2 + 1 and 127 = 63 × 2 + 1. The rule is ×2 + 1.
? = 15 × 2 + 1 = 31.
Check: 31 × 2 + 1 = 63.
31
Fractions, decimals or halving
The terms are decimals, fractions, or keep getting divided, like 81, 27, 9, 3, 1.
Check the ratio between terms first: divide one term by the next.
For fractions, write the tops and bottoms as two series.
Apply the rule once to the last term.
The same rules work on decimals and fractions. Only the numbers look different.
Find the next term: 81, 27, 9, 3, 1, ?
Show solutionHide solution
27 ÷ 81 = 1/3, and 9 ÷ 27 = 1/3.
Each term is the last one ÷ 3.
.
Two missing terms: pick the right pair
Two '?' marks sit in one series, and each option is a pair of numbers.
Find the rule from the visible terms.
Fill the first blank.
Use the rule again to fill the second blank.
Check the terms after both blanks.
Each blank follows from the one before it, so fill them in order.
Fill the blanks: 1, 2, 4, ?, 11, ?, 22
Show solutionHide solution
The first gaps are 1 and 2. Guess that the gaps rise by 1: 1, 2, 3, 4, 5, 6.
First blank: 4 + 3 = 7. Check: 7 + 4 = 11.
Second blank: 11 + 5 = 16. Check: 16 + 6 = 22.
7 and 16
Find the first term
The '?' is at the start, like ?, 12, 20, 30, 42.
Write the gaps between the visible terms.
Extend the gap row backwards by one step.
Subtract that gap from the first visible term.
A series can be read backwards just like forwards.
Find the first term: ?, 12, 20, 30, 42
Show solutionHide solution
Gaps between the visible terms: 8, 10, 12. They rise by 2.
Going back, the gap before 8 is 6.
First term: 12 − 6 = 6. Check: 6, 12, 20, 30, 42 has gaps 6, 8, 10, 12.
6
Formula sheet
Write these gaps under the series. If they are not clear, take gaps of the gaps.
d is the equal gap. Use it when the first row of gaps is constant.
Shortcuts that save time
Write the gaps under the series. If they are not equal, write the gaps of the gaps. Stop when a row is clear. Then work back up.
Find the next term: 4, 7, 12, 19, 28, ?
Show solutionHide solution
Gaps: 3, 5, 7, 9.
The gaps rise by 2, so the next gap is 11.
28 + 11 = 39.
39
Before you calculate, say the rule in one plain sentence, like 'add 3, then add 6, then add 9'. If you cannot say it in a sentence, the rule is wrong.
Find the next term: 5, 8, 14, 23, 35, ?
Show solutionHide solution
Gaps: 3, 6, 9, 12.
Rule in words: the gap grows by 3 each time.
The next gap is 15. So 35 + 15 = 50.
50
Mistakes to avoid
Where most students lose marks on this subtopic.
Guessing the rule from the last two terms only.
The rule must fit every term. Test it from the first term.
Stopping after one row of gaps when it is not equal.
Take gaps of the gaps. The second row often shows the rule.
Filling a middle blank without checking the next term.
Check that the term after the blank comes from your answer.
Reading decimals or fractions as unusual rules.
Use the same ideas. Decimals follow ×k or ÷k. Fractions split into tops and bottoms.
Trying squares first on every series.
Look at the size first. Slow growth means gaps. Fast growth means ratio.
Quick revision
Read this the night before the exam.
Look at the size first: slow growth adds, fast growth multiplies, up and down means two series.
Difference ladder: first row of gaps, then the gaps of the gaps.
If the ladder fails, try ratio, then alternate terms, then squares and cubes, then mixed steps.
Missing term: check with the term before it and the term after it.
Fractions: tops and bottoms are two separate series.
The rule must fit every visible term.
Practice: 12 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 12 questions
Suggested time 7 min · wrong answers go to your mistake notebook automatically.