Number series questions, with answers

The questions, with answers

1. 1.What is the order in which you should test a series?

   Write the differences between consecutive terms first. If they are constant, it is arithmetic; if the differences themselves change by a constant, take the second differences; if the differences look like a known sequence (primes, squares, doubling), you have found it. If differences do not help, take the ratios — a constant ratio means geometric, a growing ratio suggests multiplication by increasing numbers. Then check whether the terms are near squares or cubes (26 is 25 + 1, 63 is 64 - 1). If nothing fits, split the series into alternate terms — positions 1, 3, 5 and positions 2, 4, 6 — and test each half. Most placement series fall to the first two steps; running them in order takes twenty seconds.

2. 2.How do you solve a series with a constant or changing difference?

   For 4, 9, 14, 19, 24, ? the differences are 5, 5, 5, 5, so the next term is 29. For 3, 7, 13, 21, 31, ? the differences are 4, 6, 8, 10 — themselves growing by 2 — so the next difference is 12 and the term is 43; that is a quadratic series, and n² + n + 1 generates it. For 1, 2, 6, 15, 31, ? the differences are 1, 4, 9, 16 — the squares — so add 25 for 56. The habit that saves time is writing the differences on the paper in a row under the series rather than computing them in your head, because the second-level pattern is only visible once the first level is written down.

3. 3.How do you recognise a series built on multiplication?

   Take the ratio of consecutive terms. 5, 15, 45, 135, ? has a constant ratio of 3, so the next term is 405. 2, 6, 24, 120, ? has ratios 3, 4, 5 — multiplication by increasing numbers, the factorial series — so multiply by 6 for 720. Then there are the mixed rules, where each term is the previous one transformed by a multiply-and-add: 3, 7, 15, 31, 63, ? doubles and adds 1 each time, so the next is 127; 4, 9, 19, 39, ? doubles and adds 1 too. When the ratio is close to 2 or 3 but not exact, test double-plus-something and triple-plus-something before anything else.

4. 4.How do squares, cubes and primes appear in a series?

   Usually offset by a constant, or as differences. 2, 5, 10, 17, 26, ? is n² + 1, so the next term is 37. 0, 7, 26, 63, 124, ? is n³ - 1, so 215. A series that is exactly the squares or cubes is too easy, so setters shift them: 4, 16, 36, 64, ? is the squares of even numbers, 100 next. Primes appear either directly — 2, 3, 5, 7, 11, 13, ? gives 17 — or as differences, and a series like 3, 5, 9, 15, 23, 33 with differences 2, 4, 6, 8, 10 is easily confused with a prime-based one, which is why you write the differences down before naming the pattern. Memorise squares to 25 and cubes to 12; recognising 169 or 343 on sight is worth a question per paper.

5. 5.What is an alternating series, and how do you spot one?

   Two independent series interleaved, so consecutive differences look chaotic while every second difference is clean. 2, 10, 4, 20, 6, 30, ? — the odd positions run 2, 4, 6 and the even positions run 10, 20, 30, so the next term is 8. 1, 4, 9, 8, 25, 12, ? — odd positions are the squares 1, 9, 25 and even positions are 4, 8, 12, so the answer is 49. The tell is a series that goes up and down, or whose differences alternate in sign or size. Split it, solve each half separately, and check which half the missing position belongs to; the most common error is answering with the next term of the wrong half.

6. 6.How do you find the wrong term in a series?

   Establish the rule from the terms that agree with each other, then find the one that breaks it. In 2, 5, 10, 17, 26, 38, 50, the differences 3, 5, 7, 9, 12, 12 stop being consecutive odd numbers at 38: the rule is n² + 1, so 37 belongs there, and 38 is the wrong term (50 = 7² + 1 is correct). The trap is fixing the wrong number: 38 breaks the pattern, but so would the terms after it if the error were earlier, so always check that the rule reproduces every other term before you answer. Options in these questions often include the terms adjacent to the error, precisely to catch candidates who stop at the first anomaly.

7. 7.How do you handle a series where the operation itself changes, like +2, ×3, +2, ×3?

   Look at the pattern of operations rather than of numbers. 5, 7, 21, 23, 69, 71, ? alternates adding 2 and multiplying by 3, so the next is 213. 2, 3, 6, 11, 18, ? adds 1, 3, 5, 7 — the operation is the same but its operand walks up the odd numbers — so add 9 for 27. Some setters chain two operations into each step: ×2 + 1, ×2 + 2, ×2 + 3 gives 1, 3, 8, 19, 42, and then 89. The way to see these is to write the operation needed to get from each term to the next as a tiny expression, not just a difference; the sequence of expressions is the pattern.

8. 8.What should you do when you cannot see the pattern within a minute?

   Move on. Series questions are all worth the same mark, and a paper has easier ones behind the hard one. Before you leave, do two cheap checks: try the options — substituting each and seeing which one makes the differences or ratios clean often works backwards faster than solving forwards — and look at magnitude, since the answer to a doubling series cannot be close to the previous term. Mark the question, and come back only if time remains. The candidates who lose this section are not the ones who miss a hard series; they are the ones who spend four minutes on it and then rush the six easy ones after it.

How the diagnostic asks it

One question from the Aptitude bank, exactly as a sitting would show it. The bank has 4 on number series and 60 across Aptitude.