What should come in place of ? in the following series ?
117, 98, 80, 64, ?, 42

Question

UnlockIAS · Accepted Answer

Correct — A, 51. Take first differences along the printed terms: 117 − 98 = 19, 98 − 80 = 18, 80 − 64 = 16. Those gaps are not constant, so it is not an arithmetic series; but look at how the gaps themselves change — 19 to 18 is a fall of 1, and 18 to 16 is a fall of 2. The differences of the differences are −1, −2, and the rule continues −3, −4. That gives the remaining gaps as 16 − 3 = 13 and 13 − 4 = 9. So the missing term is 64 − 13 = 51, and the term after it is 51 − 9 = 42. The 42 is the whole point of the question. BPSC has printed the last term rather than the missing one, so the series carries its own check digit: only a rule that lands exactly on 42 can be right, and 51 is the only option that does. Test the others and each one breaks that check. Choosing 55 forces gaps of 9 and then 13 — the differences would have to shrink, shrink, shrink and then grow again, which no single rule produces. Choosing 46 forces gaps of 18 and then 4, a jump of 14 in one step where every earlier step changed by 1 or 2. Choosing 48 forces gaps of 16 and then 6, which repeats the earlier gap of 16 and then collapses by 10. Only 51 makes both the step into the blank and the step out of it obey one consistent rule, so (a) is the answer.

Why the other options are wrong — (b) The trap for a candidate who spots that the gaps are falling but not by how much. 55 sits a plausible-looking 9 below 64, and 9 is indeed a gap that appears in this series — but it is the last gap, not the fourth one. Putting 55 in the blank leaves a gap of 13 from 55 to 42, i.e. the differences would run 19, 18, 16, 9, 13, rising again at the end. A series whose gaps shrink and then grow has no single rule.  (c) Comes from mis-reading the fall in the gaps as steady rather than accelerating — treating the next gap as 18 again gives 64 − 18 = 46. But that leaves only 4 to cover from 46 to the printed 42, against a previous gap of 18. The printed final term is exactly what exposes this reading.  (d) Comes from repeating the last computed gap of 16, giving 64 − 16 = 48. That ignores the fact that the gaps have already fallen twice, and it leaves a gap of only 6 into the printed 42 — a drop of 10 in a single step, where the pattern has never changed by more than 2.

Concept: A number series question is solved by the method of finite differences, and the procedure is mechanical rather than inspired. Write the gaps between consecutive terms on a second line. If that line is constant, the series is arithmetic and you are done. If it is not, write the gaps of the gaps on a third line and look again. Most examination series resolve by the second or third line, because setters build them from low-order rules: constant difference, constant ratio, a difference that itself moves in a simple pattern, or terms generated by squares, cubes, primes or products of consecutive integers. When the differences fail to settle, switch strategies — test for a multiplicative rule by dividing consecutive terms, look for alternate-term series interleaved with one another, or check the terms against n², n³, n(n+1) and their small offsets.

State PCS 2025 Q3 — Science & Technology PYQ with Answer

State PCS 2025 Q3 — Science & Technology PYQ with Answer