BPSC 70th CCE Re-exam Prelims General Studies · Question 3 of 142

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.

• (b)55 — 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)46 — 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)48 — 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.

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.

Two habits turn these from guesswork into arithmetic. The first is to use every printed term, not just the ones before the blank. Here the blank sits fifth and 42 is printed sixth, so any candidate answer can be verified twice — once on the way in and once on the way out — and that double check kills three of the four options outright. Setters place the blank in the middle precisely to make this possible, and candidates lose the mark by computing forwards only. The second habit is to prefer the smoothest rule that fits. The gaps 19, 18, 16, 13, 9 fall by 1, 2, 3, 4 — a triangular pattern, and about as simple as a non-constant difference gets. Notice also that the whole series can be written in closed form from that: successive terms fall by 19, then by one less each time, so the sequence is 117, 98, 80, 64, 51, 42, and the next gap would be 5, giving 37. Under exam pressure the arithmetic is only three subtractions and one check; the discipline of doing the check is what separates 51 from 55.

• First differences of the printed series: 19, 18, 16, 13, 9 — falling by 1, then 2, then 3, then 4 (second differences −1, −2, −3, −4).
• The full series is therefore 117, 98, 80, 64, 51, 42, and the term after 42 would be 42 − 5 = 37.
• When a blank sits in the middle of a printed series, every candidate answer can be verified in both directions; only 51 satisfies both 64 − x = 13 and x − 42 = 9.
• Standard families a BPSC number series is built from: constant difference, constant ratio, differences in arithmetic progression, perfect squares or cubes with a small offset, products of consecutive integers such as n(n+1), and two alternating sub-series.
• Paper-I of this 70th CCE re-examination of 4 January 2025 carries 150 questions for 150 marks, with one-third of a mark deducted for every wrong answer — so a series question that can be verified in both directions is one of the few items on the paper where a candidate can mark with certainty rather than accept the negative-marking risk.

• Computing only forwards from the term before the blank and never checking against the terms printed after it — the single commonest way to lose this mark.
• Assuming a falling difference falls by a constant amount. Here it falls by 1, then 2, then 3, then 4, and reading it as a fixed −2 produces 46.
• Re-using a gap that has already appeared elsewhere in the series. The gap of 9 is real, but it belongs to the last step, not the fourth.

BPSC puts three or four pure reasoning items into General Studies Paper-I itself — number series, direction sense, clock and calendar, coding — because the state services have no separate aptitude paper at the prelims stage, so these carry the same one mark as a Bihar geography question and are the fastest marks on the paper. UPSC moved all of it into CSAT Paper-II in 2011, where it is only qualifying at 33%, which is why a UPSC aspirant preparing for BPSC has to bring the arithmetic back into the main paper's timing plan.

• practice — not a real PYQ

  What should come in place of ? in the following series ? 7, 14, 28, 56, ?, 224

  • (a)98
  • (b)112
  • (c)120
  • (d)168

  Answer(b) 112 — each term is double the one before, so 56 × 2 = 112, and the check holds because 112 × 2 = 224, the printed last term.

• practice — not a real PYQ

  What should come in place of ? in the following series ? 2, 6, 12, 20, ?, 42

  • (a)28
  • (b)30
  • (c)32
  • (d)36

  Answer(b) 30 — the terms are products of consecutive integers, 1×2, 2×3, 3×4, 4×5, 5×6, 6×7, so the missing term is 5 × 6 = 30, and 6 × 7 = 42 confirms it. Read as differences, the gaps are 4, 6, 8, 10, 12.