Series (Number & Letter)
🔒 Log in to trackWrong number in the series
🔒 Log in to trackHere the series is already complete but one term was corrupted. You must find which term breaks the rule — not what should replace it. Method: build the difference ladder as usual; the corrupted term shows up as the place where the ladder stops making sense.
Strategy: test the rule on terms 1-3 and terms 4-6 separately. The half that agrees gives you the family; the term in the other half that refuses to fit is the culprit. Verify by checking that the rule reproduces all other terms exactly.
Detailed notes
What the question asks
The series is complete, but one term was changed. You must find that wrong term — not the correct value. Think of a bank passbook with one typing mistake: every other entry follows the rule, only one breaks it.
The basic method
- Take the first three or four terms and guess the rule.
- Check the guess on every term, one by one.
- The single term that does not fit is the answer.
- Confirm: if you replace it with the correct value, the whole series must work, including the term just after it.
Why one wrong term spoils two differences
When a term is wrong, both the gap before it and the gap after it look wrong. Example: 5, 9, 13, 17, 22, 25 → differences 4, 4, 4, 5, 3. Two gaps (5 and 3) are off, and the term between them, 22, is the culprit. The correct value is 21. So look for the term that sits between two bad gaps.
Type 1: difference series
Constant or growing gaps: 3, 5, 9, 15, 23, 34, 45 → differences 2, 4, 6, 8, 11, 11. The expected gaps are 2, 4, 6, 8, 10, 12, so 34 should be 33.
Type 2: ratio and ×k ± r series
2, 6, 18, 54, 164, 486 → ×3 gives 162, not 164. For ×k ± r, calculate each term from the previous one: 5, 11, 23, 48, 95 → ×2 + 1 gives 47, not 48.
Careful: the term after the wrong one is made from the correct value. So 95 = 47 × 2 + 1 looks wrong if you use 48. Rebuild the series from the start instead of jumping between neighbours.
Type 3: squares and cubes
1, 8, 27, 64, 124, 216 → 124 should be 125 (5³). 2, 6, 12, 20, 31, 42 → n × (n+1) gives 30, not 31. Near-miss numbers like 124, 31 or 65 are strong hints.
Type 4: alternate and mixed series
Split the series into odd and even positions. 2, 10, 4, 13, 6, 16, 9, 19 → odd places 2, 4, 6, 9 (should be 8); even places 10, 13, 16, 19 are fine. For mixed operations like ×1+1, ×2+2, check each step with its own multiplier.
Type 5: wrong term in a letter series
The same idea works with letters. BD, EG, HJ, KN, NP → every cluster has a gap of 2 inside, so KN (gap 3) should be KM.
Speed tips
- The options are terms of the series. When short of time, test only the four option terms.
- Odd/even check: if all terms but one are odd, that one is suspicious — but confirm with the rule.
- The first term can also be wrong; check it against the second.
Quick revision
- The wrong term sits between two bad gaps.
- Rebuild each term from the correct previous term.
- Near-miss values (124 vs 125, 31 vs 30) are giveaways.
- Split up-down series into two series.
- Answer = the wrong term as shown, not its correct value.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Wrong term in a difference (AP / growing gap) seriesvery common3 practice Q
Gaps are almost equal or almost evenly rising, with two neighbouring gaps spoilt.
- Write the differences.
- Find the two neighbouring bad gaps.
- The term between them is wrong; confirm by putting the correct value back.
Why it works: one wrong term changes both the gap before it and the gap after it.
Example: 8, 12, 16, 21, 24, 28
Differences 4, 4, 5, 3, 4 — the bad gaps 5 and 3 sit around 21 (should be 20).
Type 2: Wrong term in a ratio or ×k ± r seriescommon3 practice Q
Terms roughly double or triple; one term is slightly off.
- Find k (and r) from the first few terms.
- Rebuild the series from the first term.
- The first term that does not match is the wrong one.
Why it works: the next term is made from the correct value, so rebuilding from the start finds the break.
Example: 3, 6, 12, 25, 48, 96
Rule ×2: 3, 6, 12, 24, 48, 96. The term 25 is wrong (should be 24).
Type 3: Wrong term in a square / cube seriescommon3 practice Q
Terms are near squares or cubes; one term misses by 1 or 2.
- Match every term with n², n³ or n(n+1) plus a fixed adjustment.
- The term that does not fit the adjustment is wrong.
Why it works: squares and cubes are exact, so a near-miss stands out.
Example: 4, 9, 16, 25, 35, 49
Squares 2² to 7²: 4, 9, 16, 25, 36, 49. 35 is wrong (should be 36).
Type 4: Wrong term in an alternate or mixed-operation seriescommon2 practice Q
The series goes up and down, or uses ×1+1, ×2+2 type steps.
- Split into odd and even places, or list the operation for each step.
- Check each small series or each step separately.
- The one term that breaks its own series is wrong.
Why it works: the error hides inside one of the two small series.
Example: 1, 20, 3, 18, 5, 17, 7
Odd places 1, 3, 5, 7 fine. Even places 20, 18, _ should be 16 → 17 is wrong.
Type 5: Wrong term in a letter / letter-cluster seriescommon2 practice Q
Letters or clusters in a series; one of them does not follow the step.
- Convert letters to positions.
- Find the step pattern (between terms and inside clusters).
- The letter or cluster that breaks it is the answer.
Why it works: it is the same wrong-term method on positions instead of numbers.
Example: C, F, I, M, O
Step +3: C, F, I, L, O. M is wrong (should be L).
Shortcut tricks
⚡ Ladder with one broken rung
Compute all differences; all but one gap will follow the pattern (constant, rising, doubling). The term touching the odd gap(s) is the wrong one.
Example: Q. 2, 5, 10, 17, 26, 38
Sol. These should be n² + 1: 2, 5, 10, 17, 26, 37. The last term 38 breaks it → 38 is wrong. (Differences 3, 5, 7, 9 are clean; the last jump is 12, not 11.)
Find the family from the clean majority; the leftover term is the answer.
Where students lose marks
Reporting the correct value (37) instead of the wrong given term (38) — read the question twice.
Choosing a rule that fits only the first three terms; always test it on the tail too.
Forgetting that exactly ONE term is wrong — if two refuse to fit, your family is wrong.
Practice sets — 14 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 7 min · wrong answers go to your mistake notebook automatically.