Which of the following numbers will replace the question mark (?) in the given series? 3, 17, 83, 329, ?, 1961

Correct — A. The differences 14, 66, 246 grow too fast to be worth chasing, so take ratios instead: 17 over 3 is about 5.7, 83 over 17 about 4.9, 329 over 83 about 4.0. The multiplier is stepping down.

Rule: ×6 − 1, ×5 − 2, ×4 − 3, ×3 − 4, ×2 − 5 — the multiplier falls by 1 and the subtraction grows by 1.

3 × 6 − 1 = 17
17 × 5 − 2 = 83
83 × 4 − 3 = 329
329 × 3 − 4 = 983 → option (a)

The term after the blank confirms the rule: 983 × 2 − 5 = 1961, exactly what the question prints.

Why the others are wrong

• (b)
  880 — 880 would need 329 × 3 − 107, a subtraction unrelated to the run 1, 2, 3. It also fails the closing check, since 880 × 2 − 5 = 1755, not 1961.
• (c)
  980 — 980 keeps the ×3 but subtracts 7 instead of 4, breaking the 1, 2, 3, 4, 5 sequence. 980 × 2 − 5 = 1955, so it misses the final term too.
• (d)
  883 — 883 needs 329 × 3 − 104. The digits deliberately echo the right answer, but the arithmetic does not: 883 × 2 − 5 = 1761.

When the terms of a series multiply rather than add, stop differencing and take ratios. Ratios that fall in an orderly way point at a multiplier that steps down by one each time.

Once you suspect a ×k − m rule, expect the subtraction to move in step with the multiplier: these series are built so that both counters march together.

A series that prints a term beyond the blank is handing you a verification. Use it. An option that reproduces the blank but fails the last term is wrong however tidy it looks.

Key facts

• The multipliers run 6, 5, 4, 3, 2 and the subtractions run 1, 2, 3, 4, 5.
• 983 × 2 − 5 = 1961, the term the question supplies after the blank.
• Falling ratios rather than a constant difference are the signature of a times-k-minus-m series.

Common traps

• Hunting for a constant difference when the terms grow this fast
• Choosing an option without testing it against the term printed after the blank

No directly related past PYQ was found.