Select the number from among the given options that can replace the question mark (?) in the following series. 625, 382, 301, 274, 265, ?

Correct — B.

Take first differences before anything else.

Rule: subtract descending powers of 3 — 3⁵, 3⁴, 3³, 3², 3¹.

625 − 243 = 382
382 − 81 = 301
301 − 27 = 274

274 − 9 = 265
265 − 3 = 262 → option (b)

Why the others are wrong

• (a)
  258 — 258 is 265 − 7, and 7 is no power of 3. The gaps run 243, 81, 27, 9, each a third of the one before, so the last gap is fixed at 3.
• (c)
  248 — 248 needs a gap of 17. The gap divides by 3 each step and the previous gap was 9, which leaves 3 and nothing else.
• (d)
  260 — 260 is 265 − 5. It comes from assuming the gaps fall by a fixed amount rather than by a fixed ratio; after 9 comes 3, not 5.

A falling series whose gaps shrink fast is almost always geometric in the gaps, not in the terms.

First differences here are −243, −81, −27, −9. Each is the previous one divided by 3, so they are 3⁵, 3⁴, 3³, 3² and the next is 3¹ = 3.

Recognising 243 as 3⁵ is the whole question. Note that the opening term 625 is 5⁴ — a coincidence the differences never use.

Key facts

• First differences: 243, 81, 27, 9 — the powers 3⁵ down to 3², each a third of the one before.
• The next gap is 3¹ = 3, so 265 − 3 = 262.
• Worth knowing on sight: 3⁵ = 243 and 5⁴ = 625.

Common traps

• Testing division on the terms themselves before taking first differences
• Reading the shrinking gaps as an arithmetic drop instead of a ratio, which produces 258 or 260

No directly related past PYQ was found.