Select the number from among the given options that can replace the question mark (?) in the following series. 389, 420, 457, 498, 541, ?

Correct — C. Take first differences before anything else.

420 − 389 = 31
457 − 420 = 37
498 − 457 = 41
541 − 498 = 43

Rule: the differences are consecutive prime numbers. After 43 the next prime is 47.

541 + 47 = 588 → option (c).

Why the others are wrong

• (a)
  572 — 572 is 541 + 31, which reuses the first difference instead of carrying the sequence forward.
• (b)
  584 — 584 is 541 + 43, repeating the last difference rather than moving on to the next prime.
• (d)
  598 — 598 needs a step of 57, and 57 is not prime — it factorises as 3 × 19.

When a series climbs unevenly, take first differences before looking for anything else. If the differences form a recognisable list, the rule lives in them and not in the terms.

Here they are 31, 37, 41 and 43. Reading those as consecutive primes needs the primes of the thirties and forties at your fingertips: 31, 37, 41, 43, 47, 53. Nothing between 31 and 37 is prime, which is why the first jump is 6 while the later ones are 4 and 2.

Key facts

• The differences of the series are 31, 37, 41 and 43, four consecutive primes.
• The prime after 43 is 47, so the next term is 541 + 47 = 588.
• 57 is not prime, since 57 = 3 × 19.

Common traps

• Assuming the differences rise by a fixed amount and adding 45 or 46
• Treating 57 as a prime number
• Adding back a difference the series has already used

No directly related past PYQ was found.