BPSC 69th CCE Prelims General Studies · Question 145 of 144

Correct — B, 182. Work the first differences of the terms the paper actually prints. 156 − 132 = 24; the next gap is the unknown one; then 210 → 240 = 30 and 240 → 272 = 32. The two gaps at the tail differ by 2, so the ladder of differences is climbing by 2 at every step, which forces the two unknown gaps to be 26 and 28. That gives 132 + 24 = 156, 156 + 26 = 182, and 182 + 28 = 210 — and 210 is a term the paper prints, so the rule is checked against the question rather than assumed from it. A second, independent reading lands on the same number: every term is the product of two consecutive integers, 11 × 12 = 132, 12 × 13 = 156, 14 × 15 = 210, 15 × 16 = 240, 16 × 17 = 272, so the missing thirteenth-and-fourteenth product is 13 × 14 = 182. Numbers of this form, n(n + 1), are called pronic or oblong numbers. There is also a one-line shortcut worth memorising for the exam hall: when the second difference is a constant d, any term lying between two known terms equals (sum of its two neighbours − d) ÷ 2. Here that is (156 + 210 − 2) ÷ 2 = 364 ÷ 2 = 182, with no ladder to write out at all.

• (a)196 — 196 is 14², and that is exactly the bait: a candidate who notices 210 = 14 × 15 and 240 = 15 × 16 and then works backwards can slip to 14 × 14 instead of 13 × 14. Test it against the printed terms and it collapses — 156 → 196 is a gap of 40 and 196 → 210 a gap of 14, so the ladder would read 24, 40, 14, 30, 32, which is no pattern at all.
• (c)199 — The only odd number on offer, and it can be struck out before any rule is found: one of two consecutive integers is always even, so every term of this series is even. 199 is also prime, which makes it the natural-looking random guess for a candidate who has run out of time. On the difference test it yields gaps of 43 and 11, nothing like the smooth climb the tail of the series shows.
• (d)204 — The trap for a candidate who takes the opening gap of 24 and assumes the gaps double: 156 + 48 = 204. It fails on the very next step, because 204 → 210 is a gap of 6 — smaller than the 24 the series opened with — while the printed tail gaps of 30 and 32 are plainly still rising. 204 factorises as 12 × 17, which fits no consecutive-integer pattern either.

A number-series item hides one rule and asks you to reconstruct it from the terms shown. The families worth recognising on sight are few: a constant difference (arithmetic), a constant ratio (geometric), a difference that itself moves in a pattern (which means the underlying rule is quadratic or higher), squares or cubes with a constant added or subtracted, products of consecutive integers, and alternating series in which the odd and even positions follow separate rules. The single most productive first move is always the same — write the first differences underneath the terms. If they are constant, the rule is linear. If they are not, take the differences of those differences. A constant second difference is the signature of a quadratic rule, and this question is exactly that case: the second difference is 2 throughout, which corresponds to the rule n² + n = n(n + 1).

The layout of this particular series is a gift, and recognising why is the transferable skill. The question mark sits at position three, which leaves three consecutive terms untouched at the tail — 210, 240, 272. Those three alone give two clean differences, 30 and 32, and therefore the whole structure of the ladder before you have looked at the gap at all. Deriving the rule from the part of the series the examiner left intact, and only then walking it back across the hole, is far safer than guessing from the two terms nearest the question mark. The discriminating observation is that the second difference is 2, not that the terms are pronic numbers — the product form is elegant and it is a useful cross-check, but a candidate who never spots 13 × 14 still reaches 182 in about fifteen seconds from the difference ladder. Two habits pay for themselves here. First, always verify a candidate rule against a term the paper prints, not only against the blank: 182 + 28 = 210 is the step that turns a guess into an answer. Second, use parity as a free filter — with every printed term even, an odd option is dead on arrival.

• The six terms are pronic (oblong) numbers n(n + 1) for n = 11 to 16: 11 × 12 = 132, 12 × 13 = 156, 13 × 14 = 182, 14 × 15 = 210, 15 × 16 = 240, 16 × 17 = 272
• First differences run 24, 26, 28, 30, 32 — an arithmetic progression with common difference 2, so the second difference is a constant 2 and the underlying rule is quadratic
• Interpolation shortcut for a constant second difference d: the missing middle term = (sum of the two neighbours − d) ÷ 2, here (156 + 210 − 2) ÷ 2 = 182
• A pronic number is always even, since one of any two consecutive integers is even, and is twice a triangular number: 182 = 2 × 91, and 91 is the 13th triangular number
• The series continues 17 × 18 = 306 and 18 × 19 = 342, the gaps being 34 and 36

• Building the rule from the two terms next to the question mark instead of from the untouched run of terms at the tail
• Accepting a rule that fits the blank without re-checking it against a term the paper actually prints
• Ignoring parity — every printed term here is even, which kills an odd option before any arithmetic is done

BPSC places a short reasoning block at the very end of the General Studies paper, and the numeric items in it are deliberately mechanical — one hidden rule, four numeric options, no wordplay — so they are among the fastest marks on the paper for a prepared candidate. UPSC keeps this family in CSAT Paper II under basic numeracy and general mental ability, where it is usually dressed as a sequence inside an applied situation rather than presented as a bare row of numbers.

No directly related past PYQ was found.

• practice — not a real PYQ

  What should come in place of the question mark (?) in the following number series? 210 240 272 ? 342

  • (a)292
  • (b)306
  • (c)300
  • (d)312

  Answer(b) 306 — the terms are n(n + 1): 14 × 15 = 210, 15 × 16 = 240, 16 × 17 = 272, 17 × 18 = 306, 18 × 19 = 342; equivalently the gaps run 30, 32, 34, 36.

• practice — not a real PYQ

  What should come in place of the question mark (?) in the following number series? 6 12 20 30 ? 56

  • (a)40
  • (b)42
  • (c)44
  • (d)46

  Answer(b) 42 — the differences are 6, 8, 10, 12, 14, so the missing term is 30 + 12 = 42; the terms are the products 2 × 3, 3 × 4, 4 × 5, 5 × 6, 6 × 7, 7 × 8.