Which of the following numbers will replace the question mark (?) in the given series? 7, 13, ?, 145, 721, 4321

Correct — D. Constant differences and a constant ratio both fail here, so test a growing multiplier.

7 × 2 − 1 = 13 ✓
13 × 3 − 2 = 37

37 × 4 − 3 = 145 ✓
145 × 5 − 4 = 721 ✓
721 × 6 − 5 = 4321 ✓

Rule: × n − (n − 1), with n running 2, 3, 4, 5, 6.

The missing term is 13 × 3 − 2 = 37, option (d).

Why the others are wrong

• (a)
  23 — 23 fits an additive reading of the first gap, +6 then +10. It fails the rule the rest of the series obeys, though: 23 × 4 − 3 = 89, not the 145 that is printed next.
• (b)
  36 — 36 is even, while 7, 13, 145, 721 and 4321 are all odd. This rule preserves oddness at every step, so no even number can fill the gap.
• (c)
  25 — 25 repeats the first step's multiplier — 13 × 2 − 1 = 25. But the multiplier grows, and 25 × 3 − 2 = 73, not the 145 that follows.

A series that outgrows any constant difference but has no constant ratio is usually driven by a changing multiplier. The standard family is × 2 − 1, × 3 − 2, × 4 − 3, with cousins × 2 + 1, × 3 + 2.

Test the family on a gap you can actually see rather than on the gap you are being asked for. Here 145 → 721 is the one to use: 145 × 5 = 725 and 725 − 4 = 721, which pins n = 5 at that step and fixes the whole pattern.

Working from a known consecutive pair is both faster and safer than working from the missing term.

Key facts

• The rule is next = current × n − (n − 1), with n taking 2, 3, 4, 5 and 6 down the series.
• Written the other way, each term is n × (previous − 1) + 1 — for instance 3 × (13 − 1) + 1 = 37.
• Every term of this series is odd, which rules out an even option before any arithmetic.

Common traps

• Chasing first and second differences on a series that multiplies.
• Fixing the multiplier at 2 or 3 instead of letting it grow one step at a time.

No directly related past PYQ was found.