Number Series

Number Series — Summary

Number series is the most common Logical Reasoning question in placements (TCS, Infosys, Wipro) — expect 2 to 4 per paper. You are given a sequence and must find the next term or the wrong term, so recognising the underlying rule fast is everything.

Common patterns to scan for

Arithmetic: constant difference (3, 7, 11, 15 → +4).
Geometric: constant ratio (3, 6, 12, 24 → ×2).
Squares/cubes: 1, 4, 9, 16 (n²); 1, 8, 27, 64 (n³).
Difference-of-differences: 2, 6, 12, 20, 30 → gaps 4, 6, 8, 10 → next 42.
Primes: 2, 3, 5, 7, 11, 13, 17.
Mixed / alternating: two interleaved series.

Series	Rule	Next
3, 6, 12, 24	×2	48
2, 6, 12, 20, 30	n²+n	42
1, 4, 9, 16, 25	squares	36
2, 3, 5, 7, 11	primes	13

Exam Tricks & Tips

🎯 First check the DIFFERENCES; if they're constant it's arithmetic, if they grow steadily try a second difference.
🎯 If terms grow fast, test RATIOS for a geometric or ×n pattern.
🎯 Watch for squares/cubes ±1 (e.g. 2, 5, 10, 17 = n²+1).
🎯 Alternating series: separate the odd-position and even-position terms.
🎯 For "find the wrong term", predict each term and spot the mismatch.
❌ Don't force one rule — if differences and ratios both fail, look for interleaving or squares/cubes.

Expected exam pattern: find the next term, the missing middle term, or the odd/wrong term. 2 to 4 questions, ~40 seconds each.

Quick recap: scan differences, then ratios, then squares/cubes/primes, then interleaving; predict and verify to catch the wrong term.

Number Series — Flashcards

Q1. First thing to check in a number series?
A1. The differences between consecutive terms.

Q2. Next term: 3, 6, 12, 24, …?
A2. 48 (ratio ×2).

Q3. Next term: 2, 6, 12, 20, 30, …?
A3. 42 (gaps 4,6,8,10,12; also n²+n).

Q4. Rule for 1, 4, 9, 16, 25?
A4. Perfect squares; next is 36.

Q5. Next prime after 2, 3, 5, 7, 11, 13?
A5. 17.

Q6. When do you check ratios instead of differences?
A6. When the terms grow quickly (multiplicative pattern).

Q7. Identify 2, 5, 10, 17, 26?
A7. n² + 1 → next 37.

Q8. Handle an alternating series how?
A8. Split into odd-position and even-position sub-series.

Q9. Next term: 1, 8, 27, 64, …?
A9. 125 (cubes).

Q10. "Wrong term" approach?
A10. Predict each term from the rule and flag the mismatch.

Q11. Next term: 5, 11, 23, 47, …?
A11. 95 (each ×2 + 1).

Q12. Series 4, 9, 16, 25, 36 — rule?
A12. Squares of 2,3,4,5,6.

Number series is a pattern-hunting sport: the setter hides a rule, you find it in under 30 seconds. The trick is not cleverness but a fixed checklist you run in order, so you never stare blankly at the numbers.

What this topic tests: recognising the arithmetic/geometric/mixed rule behind a sequence and finding the missing or wrong term.

The method

Beginner — the difference-first checklist

Run these in order until one fits:

Constant difference (add/subtract same number): 3, 7, 11, 15 (+4).
Constant ratio (multiply/divide): 3, 6, 12, 24 (×2).
Growing difference — look at the differences themselves: 2, 3, 5, 8, 12 (diffs 1,2,3,4).

Intermediate — the special families

Squares/cubes (±k): 2, 5, 10, 17 (n²+1); 1, 8, 27, 64 (cubes).
Primes: 2, 3, 5, 7, 11, 13.
Alternating / two interleaved series: 1, 4, 2, 8, 3, 12 → odd positions 1,2,3 and even positions ×4.
×n then +/−: 2, 5, 11, 23 (×2 then +1).

Advanced — the fast approach

Compute the first differences; if not constant, compute the second differences. Constant second differences ⇒ a square/quadratic rule.
If the numbers jump wildly, suspect multiply-based or interleaved series and check alternate terms.
For "find the wrong term", assume the simplest rule and spot the single number that breaks it.

Worked example

Find the next term: 4, 9, 19, 39, 79, ?
Rule test: each term ×2 then +1 → 4×2+1 = 9, 9×2+1 = 19, 19×2+1 = 39, 39×2+1 = 79. So next = 79×2+1 = 159. (Differences 5,10,20,40 doubling also confirm it.)

Where it appears

TCS NQT, Infosys, Wipro, Cognizant, Capgemini reasoning; a guaranteed cluster, often the fastest marks in the section.

Speed tricks and shortcuts

Always take differences first — it settles most series in one step.
Wild jumps ⇒ think ×n±k or two interleaved sequences.
Keep the squares to 20² and cubes to 12³ memorised for instant recognition.
Mnemonic: "Differences, ratios, squares, interleave — run the list, don't guess."

❌ Common mistake

Forcing a single rule when the series is actually two interleaved sequences. If nothing works and terms zig-zag, split into alternate positions and check each half separately.

🔑 Key points

✓- Checklist order: constant difference → ratio → growing difference → squares/cubes → interleaved.
✓- Second differences constant ⇒ quadratic (square-based) rule.
✓- Watch for ×n±k patterns behind doubling differences.
✓- Memorise squares/cubes/primes for fast spotting.

🔁 Recap

✓Run a fixed checklist — differences, then ratios, then square/cube and interleaving — instead of guessing. First and second differences crack the large majority of number series quickly.

Number Series — Worked Example

Worked Example

Problem: Find the next term in the series: 2, 6, 12, 20, 30, ?

Solution: When a series has no obvious ratio, examine the differences between consecutive terms.

6 − 2 = 4
12 − 6 = 6
20 − 12 = 8
30 − 20 = 10

The differences are 4, 6, 8, 10 — increasing by 2 each time. So the next difference is 12.

Next term = 30 + 12 = 42.

Cross-check with a closed form: the terms are 1×2, 2×3, 3×4, 4×5, 5×6, so the pattern is n(n+1). The 6th term is 6×7 = 42. ✓

Answer: 42.