Number series for SSC CHSL

Every number series has a rule, and the rule is almost always one of a short list. Candidates who lose time on series are usually not missing some clever idea. They are testing rules in a random order, abandoning each one halfway, and coming back to it later.

The fix is to check the rules in the same order every time. Most series fall to the first two checks. The rest fall within the next three, and you rarely need more than 30 seconds.

How it is asked

SSC CHSL asks number series in three forms:

• Next term: "7, 10, 16, 25, 37, ?"
• Missing term: "3, 7, ?, 31, 63"
• Wrong term: "Find the number that does not fit: 5, 10, 20, 40, 82, 160"

The options matter. A series can sometimes be continued in two ways, and the options are there to settle which rule the setter used. Check your answer against the options before you commit.

The checking order

1. Differences. Subtract each term from the next. Constant differences give an arithmetic series; differences that grow by a fixed amount give a second-level series.
2. Differences of differences. If the first differences are 3, 6, 9, 12, their differences are 3, 3, 3.
3. Ratios. If the terms grow fast, divide: × 2, × 3, or × 2, × 3, × 4.
4. Squares and cubes. Compare each term with n², n³, n² ± 1 or n³ ± 1.
5. Mixed operations. × 2 + 1, × 3 − 2, and so on. A hint: if the differences themselves double, try × 2 plus a constant.
6. Alternating series. Split odd and even positions and treat them as two series.
7. Primes and sums. Consecutive primes; each term the sum of the two before.

Why this order: differences are the cheapest check and catch the most series. Ratios catch the fast growers. Squares and cubes explain series whose differences look irregular but grow smoothly. Alternating series are checked late because they only show themselves when nothing else works.

The common patterns

Learn squares to 30 and cubes to 15. Recognising 2, 9, 28, 65, 126 as "one more than the cubes" is instant if you know the cubes and slow if you don't.

Missing and wrong terms

For a missing term, find the rule from the terms on either side of the gap. Use the terms after the gap as a check, not just the ones before it.

For a wrong term, the correct answer is the one number whose change fixes the whole series. Find a rule that fits most of the terms, then correct the term that breaks it and confirm every later term follows. If a series goes 40, 82, 160, changing 82 to 80 fixes both steps. Changing 160 would still leave 40 → 82 broken.

Six worked questions

Q1. 7, 10, 16, 25, 37, ? (a) 50 (b) 52 (c) 54 (d) 55
Differences: 3, 6, 9, 12. The next difference is 15, so 37 + 15 = 52. Answer: (b) 52.

Q2. 2, 5, 11, 23, 47, ? (a) 94 (b) 97 (c) 96 (d) 95
Differences 3, 6, 12, 24 double each time, which suggests × 2 plus a constant. Each term is × 2 + 1: 47 × 2 + 1 = 95. Check with differences: the next difference is 48, and 47 + 48 = 95. Answer: (d) 95.

Q3. 2, 9, 28, 65, 126, ? (a) 217 (b) 216 (c) 215 (d) 218
The terms are one more than the cubes: 1³ + 1, 2³ + 1, 3³ + 1, 4³ + 1, 5³ + 1. Next is 6³ + 1 = 217. The differences (7, 19, 37, 61, 91) and their differences (12, 18, 24, 30) agree. Answer: (a) 217.

Q4. 2, 5, 4, 10, 8, 20, 16, ? (a) 24 (b) 32 (c) 40 (d) 48
Differences go up and down, so split the series. Odd positions: 2, 4, 8, 16 (× 2). Even positions: 5, 10, 20 (× 2). The missing term is the 8th, an even position, so 20 × 2 = 40. Answer: (c) 40.

Q5. 3, 7, ?, 31, 63 (a) 13 (b) 15 (c) 14 (d) 17
7 = 3 × 2 + 1, and 63 = 31 × 2 + 1. So the gap is 7 × 2 + 1 = 15, and 15 × 2 + 1 = 31 confirms it. Answer: (b) 15.

Q6. Find the wrong term: 5, 10, 20, 40, 82, 160. (a) 10 (b) 40 (c) 82 (d) 160
The rule is × 2. 40 × 2 = 80, not 82, and 80 × 2 = 160 fits the next term. Changing 82 alone fixes the series. Answer: (c) 82.

Common traps

Practice

1. 6, 11, 21, 36, 56, ? (a) 81 (b) 79 (c) 76 (d) 86
2. 4, 6, 10, 18, 34, ? (a) 50 (b) 64 (c) 68 (d) 66
3. 1, 8, 27, 64, ? (a) 100 (b) 125 (c) 128 (d) 216
4. 3, 4, 7, 11, 18, 29, ? (a) 47 (b) 45 (c) 40 (d) 49
5. Find the wrong term: 2, 6, 12, 20, 31, 42. (a) 6 (b) 20 (c) 31 (d) 42
6. 1, 3, 2, 6, 4, 12, 8, ? (a) 16 (b) 18 (c) 36 (d) 24
7. 720, 360, 120, 30, 6, ? (a) 0 (b) 2 (c) 1 (d) 3
8. 10, 18, ?, 66, 130 (a) 34 (b) 32 (c) 30 (d) 36

Answers:

1. (a) 81. Differences 5, 10, 15, 20, so the next is 25: 56 + 25 = 81.
2. (d) 66. Differences 2, 4, 8, 16 double, so the next is 32: 34 + 32 = 66. Equivalently × 2 − 2.
3. (b) 125. Cubes: 5³ = 125.
4. (a) 47. Each term is the sum of the two before: 18 + 29 = 47.
5. (c) 31. The terms are n(n + 1): 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6 = 30, 6 × 7 = 42. Changing 31 to 30 gives differences 4, 6, 8, 10, 12.
6. (d) 24. Odd positions 1, 2, 4, 8; even positions 3, 6, 12. The 8th term is an even position: 12 × 2 = 24.
7. (c) 1. ÷ 2, ÷ 3, ÷ 4, ÷ 5, so next ÷ 6: 6 ÷ 6 = 1.
8. (a) 34. Each term is × 2 − 2: 18 × 2 − 2 = 34, and 34 × 2 − 2 = 66 confirms it.

What to do next

• Write out squares to 30 and cubes to 15 from memory until you make no errors.
• Solve 20 series a day for a week, writing the check number (1 to 7) that cracked each one.
• Mix in five "wrong term" questions a day; they are slower and worth rehearsing.
• Letter series use the same ideas with positions instead of numbers, so continue with the alphabet test, and use odd one out to practise number properties.