Number Series: Missing Term, Wrong Term and Mixed Rules
Number Series: Missing Term, Wrong Term and Mixed Rules
A number series hides one rule; your job is to find it fast. Always start with the differences between consecutive terms. If they show no pattern, check ratios, squares or cubes, then the "multiply and add" family, and finally two series interleaved in alternate places. NTPC asks the missing term, the wrong term, and fast-growing multiply-add series.
1. Rule-Finding Order
| Step | Check | Sign that it fits |
|---|---|---|
| 1 | First differences | Constant, or rising by a fixed step |
| 2 | Second differences | Constant (the first differences form an AP) |
| 3 | Ratios | Terms roughly double or triple |
| 4 | Squares and cubes | Terms sit next to 1, 4, 9, 16 ... or 1, 8, 27, 64 ... |
| 5 | Multiply and add | Fast growth: × n + n, × 2 + 1, × n + n² |
| 6 | Alternate terms | Odd and even places follow separate rules |
Differences that are squares (1, 4, 9, 16) or primes (2, 3, 5, 7, 11) are common
Squares near the terms: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Cubes near the terms: 125, 216, 343, 512, 729, 1000
✗ 3, 6, 12, 24, 46, 96: the missing term after 96 is 192 | ✓ First check for a wrong term: 46 breaks the doubling (it should be 48)
✗ Every series has one rule for all terms | ✓ In alternate series, odd and even places follow different rules
हिंदी नोट: संख्या श्रेणी में पहले क्रमागत पदों का अंतर निकालिए। अंतर से नियम न मिले तो अनुपात, वर्ग-घन, "गुणा करके जोड़" और एकांतर पद देखिए।
Exam Pointer: Verified NTPC pattern: a fast-growing multiply-and-add series with the next term missing (August 2025 UG CBT-1). Difference, square-cube, wrong-term and alternating series had no verified NTPC shift in our check and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Difference-based series
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the next term: 5, 11, 19, 29, 41, ?
Differences 6, 8, 10, 12 rise by 2, so the next difference is 14: 41 + 14 = 55.
Answer: 55
EXAMATLAS LEVEL
Q. Find the next term: 3, 4, 8, 17, 33, ?
Differences 1, 4, 9, 16 are squares, so the next is 25: 33 + 25 = 58.
Answer: 58
Pattern 2: Multiply-and-add series (fast growth)
[PYQ: NTPC UG CBT-1 29-Aug-2025 Shift-3]
EXAM LEVEL
Q. Find the next term: 2, 5, 11, 23, 47, ?
Each term is the previous × 2 + 1: 47 × 2 + 1 = 95.
Answer: 95
EXAMATLAS LEVEL
Q. Find the next term: 4, 5, 12, 39, 160, ?
4 × 1 + 1 = 5, 5 × 2 + 2 = 12, 12 × 3 + 3 = 39, 39 × 4 + 4 = 160, so the next is 160 × 5 + 5 = 805.
Answer: 805
Pattern 3: Squares and cubes
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the next term: 2, 5, 10, 17, 26, ?
The terms are n² + 1 for n = 1 to 5, so the next is 6² + 1 = 37.
Answer: 37
EXAMATLAS LEVEL
Q. Find the next two terms: 0, 7, 26, 63, 124, ?, ?
The terms are n³ − 1 for n = 1 to 5, so the next two are 216 − 1 = 215 and 343 − 1 = 342.
Answer: 215 and 342
Pattern 4: Wrong term in a series
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the wrong term: 3, 6, 12, 24, 46, 96.
The rule is doubling: 24 × 2 = 48, and 48 × 2 = 96 confirms it. So 46 is wrong.
Answer: 46
EXAMATLAS LEVEL
Q. Find the wrong term: 2, 3, 10, 39, 170, 885.
The rule is × n + n²: 2 × 1 + 1 = 3, 3 × 2 + 4 = 10, 10 × 3 + 9 = 39, 39 × 4 + 16 = 172, and 172 × 5 + 25 = 885 confirms 172. So 170 is wrong.
Answer: 170
CBT-2 LEVEL
Pattern 5: Two series in alternate places
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the next term: 1, 10, 3, 9, 5, 8, 7, ?
Odd places: 1, 3, 5, 7 (+2). Even places: 10, 9, 8 (−1). The next even-place term is 7.
Answer: 7
EXAMATLAS LEVEL
Q. Find the next term: 3, 20, 9, 17, 27, 14, 81, ?
Odd places: 3, 9, 27, 81 (× 3). Even places: 20, 17, 14 (−3). The next term is 11.
Answer: 11
60-Second Revision
- Differences first, then second differences.
- Fast growth means multiply-and-add: × n + n, × 2 + 1, × n + n².
- Spot squares and cubes ±1.
- For a wrong term, confirm the rule with the terms after it.
- Alternate places can follow two different rules.