Find the next number in the series:3, 7, 15, 31, 63, ?

Correct — A.
Rule: each term = previous × 2 + 1.

3 × 2 + 1 = 7, 7 × 2 + 1 = 15
15 × 2 + 1 = 31, 31 × 2 + 1 = 63
63 × 2 + 1 = 127 → option (a).

Check: the gaps 4, 8, 16, 32 double, so the next gap is 64, and 63 + 64 = 127.

Why the others are wrong

• (b)
  125 — 125 is 63 × 2 − 1: it subtracts the 1 the rule adds. Each given step doubles and then adds 1, as 31 × 2 + 1 = 63 shows.
• (c)
  126 — 126 is 63 × 2, doubling without the + 1. Each given step adds 1 after doubling, as 15 × 2 + 1 = 31 shows.
• (d)
  129 — 129 needs a gap of 66 after 63. The gaps run 4, 8, 16, 32, each double the last, so the next gap is 64.

Two readings of this series agree. Doubling plus one: each term is twice the one before, plus 1. Doubling gaps: the differences run 4, 8, 16, 32, so the next is 64.

The terms are also each one less than a power of 2: 3 = 2² − 1, 7 = 2³ − 1, up to 63 = 2⁶ − 1, so the next is 2⁷ − 1 = 127.

Key facts

• Rule here: × 2 + 1, with gaps 4, 8, 16, 32, 64.
• One less than powers of 2: 3, 7, 15, 31, 63, 127, 255.
• A × 2 + c series always has doubling gaps.

Common traps

• Doubling 63 and forgetting the + 1, which gives 126
• Flipping the constant to − 1 on the last step, which gives 125

No directly related past PYQ was found.