Number series for RRB NTPC: a checking order, patterns and practice

Number series questions give you a run of numbers and ask for the next one, a missing one in the middle, or the one that does not fit. They look endless in variety, but almost every NTPC series is built from a handful of rules. If you check those rules in the same order every time, most series open up in 20–30 seconds.

What slows candidates down is not the maths. It is trying rules at random, or accepting a rule that fits the first three terms and fails on the fourth.

A checking order

1. Differences. Subtract each term from the next. Are the differences constant, or do they form their own pattern (2, 4, 6, 8 or 1, 4, 9, 16)?
2. Ratios. Is each term a multiple of the one before (× 2, × 3), or does the multiplier change (× 1, × 2, × 3)?
3. Squares and cubes. Are the terms at or near 1, 4, 9, 16… or 1, 8, 27, 64…?
4. Primes. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
5. Combined rules. × 2 + 1, × 3 − 1, n² + n, n³ − 1.
6. Alternating series. Two series interleaved: one in the odd places, one in the even places.

How to tell which to try first: if the numbers grow slowly, start with differences. If they grow fast (roughly doubling or more), try ratios or combined rules. If the series goes up and down, it is almost certainly alternating.

Patterns at a glance

Squares and cubes to know

Worked questions

Q1. 2, 6, 12, 20, 30, ?

Differences: 4, 6, 8, 10. The next difference is 12, so the answer is 42. Cross-check: the terms are 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, so the next is 6 × 7 = 42.

Q2. 5, 11, 23, 47, ?

The terms roughly double, so try a combined rule. 5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47. Answer: 47 × 2 + 1 = 95.

Q3. 3, 4, 8, 17, 33, ?

Differences: 1, 4, 9, 16. These are squares, so the next difference is 25. Answer: 33 + 25 = 58.

Q4. 1, 2, 6, 24, 120, ?

Ratios: × 2, × 3, × 4, × 5. The next multiplier is × 6. Answer: 720.

Q5. 1, 10, 3, 20, 5, 30, ?

The series rises and falls, so split it. Odd places: 1, 3, 5. Even places: 10, 20, 30. The missing term is the 7th, an odd place, so it continues 1, 3, 5: answer 7.

Q6. 0, 7, 26, 63, 124, ?

Each term is one less than a cube: 1 − 1, 8 − 1, 27 − 1, 64 − 1, 125 − 1. Answer: 216 − 1 = 215.

Q7. Wrong term. 4, 9, 16, 25, 35, 49

These should be consecutive squares: 2², 3², 4², 5², 6², 7². 6² is 36, not 35. The wrong term is 35.

Finding the wrong term, step by step

1. Look at the differences and find where the pattern breaks.
2. A single wrong term usually disturbs two differences: the one before it and the one after it.
3. Guess the correct value of that term, and check that both differences now fit.
4. The term you changed is the answer.

For example, in 2, 5, 10, 17, 27, 37 the differences are 3, 5, 7, 10, 10. The pattern should be 3, 5, 7, 9, 11. Changing 27 to 26 gives differences 9 and 11. So 27 is wrong.

Common mistakes

• Checking only the first three terms. A rule must fit every term, including the last one given.
• Missing an alternating series. If the numbers jump up and down, split them before doing anything else.
• Changing a correct term in wrong-term questions. Fix the rule first, then find the term that breaks it.
• Stopping at first differences. If the differences are not constant, take the differences of the differences.
• Arithmetic slips in combined rules. Write each step (× 2, then + 1) rather than doing both in your head.

Practice set

1. 7, 14, 28, 56, ?
2. 3, 5, 9, 15, 23, ?
3. 4, 9, 19, 39, ?
4. 6, 11, 21, 36, 56, ?
5. 2, 12, 36, 80, 150, ?
6. 5, 8, 10, 16, 15, 24, 20, ?
7. 720, 360, 120, 30, 6, ?
8. Find the wrong term: 2, 5, 10, 17, 27, 37

Answers:

1. 112. Each term × 2.
2. 33. Differences 2, 4, 6, 8, so the next is + 10.
3. 79. Each term × 2 + 1.
4. 81. Differences 5, 10, 15, 20, so the next is + 25.
5. 252. The terms are n³ + n²: 1 + 1, 8 + 4, 27 + 9, 64 + 16, 125 + 25, then 216 + 36.
6. 32. Odd places 5, 10, 15, 20; even places 8, 16, 24, 32. The missing term is in an even place.
7. 1. Divide by 2, 3, 4, 5, then 6.
8. 27. The series is n² + 1: 2, 5, 10, 17, 26, 37.

What to do next

• Learn the squares table to 20 and cubes to 10 until you recognise them on sight.
• For 20 series, write the differences under the terms before you try anything else.
• Move on to letter series, which are number series in disguise.
• Revisit odd one out for number properties like primes and cubes.