Select the number from among the given options that can replace the question mark (?) in the following series. 45, 52, 63, 76, 93, ?

Correct — D. Take first differences down the row.

52 − 45 = 7
63 − 52 = 11
76 − 63 = 13
93 − 76 = 17

Rule: the gaps are consecutive prime numbers — 7, 11, 13, 17 — so the next gap is 19.

93 + 19 = 112 → option (d).

Why the others are wrong

• (a)
  116 — 116 needs a gap of 23. That is the prime after 19, so it skips a term of the difference sequence.
• (b)
  118 — 118 needs a gap of 25, which is 5 × 5 and not prime at all, so it breaks the pattern the first four gaps set up.
• (c)
  122 — 122 needs a gap of 29, two primes past 19.

When a series has no obvious ratio, write the first differences underneath it. If those differences are themselves a recognisable sequence, the series is solved.

Here the differences are the primes from 7 upward. The same answer arrives by a second route: the second differences run 4, 2, 4, so the next gap is 17 + 2 = 19.

Two independent readings landing on the same value is the strongest confirmation available in a series question.

Key facts

• The completed series is 45, 52, 63, 76, 93, 112.
• Its gaps are 7, 11, 13, 17, 19 — the consecutive primes starting at 7.
• The second differences are 4, 2, 4, 2, which reaches the same next gap of 19.

Common traps

• Counting 9, 15 or 21 as prime and inserting one of them into the gap sequence
• Repeating the last gap instead of advancing it, which lands on 110 and is not on the option list
• Looking for a multiplicative rule, when 52/45 and 63/52 are not equal so no constant ratio exists

No directly related past PYQ was found.