What will come at the place of question mark? 1, 3, 7, 15, 31, ?

Correct — B.
Rule: double the term, then add 1. The differences, 2, 4, 8, 16, double each time.

1 × 2 + 1 = 3
3 × 2 + 1 = 7
7 × 2 + 1 = 15

15 × 2 + 1 = 31
31 × 2 + 1 = 63 → option (b).

Why the others are wrong

• (a)
  62 — 62 is 31 × 2 without the +1. Every term so far is odd (1, 3, 7, 15, 31), and doubling then adding 1 keeps it odd.
• (c)
  64 — 64 is 2⁶, a power of two. The terms sit one below the powers of two, 2 − 1, 4 − 1 … 32 − 1, so the next is 64 − 1.
• (d)
  65 — 65 adds 34 to 31. The differences double each time, 2, 4, 8, 16, so the next one is 32, giving 63.

Number series reward checking the differences first. Here they are 2, 4, 8, 16: each difference is double the last, so the next is 32, and 31 + 32 = 63.

The same series has a second description: each term is one less than a power of two (2 − 1, 4 − 1, 8 − 1, 16 − 1, 32 − 1). Both readings give 64 − 1 = 63, a quick check that the rule is right.

Key facts

• 1, 3, 7, 15, 31, 63 are 2¹ − 1 up to 2⁶ − 1.
• Rule in words: double the term and add 1.
• The differences 2, 4, 8, 16, 32 double each time.

Common traps

• Doubling without adding 1, which gives 62
• Jumping to the power of two, 64, instead of one less than it

No directly related past PYQ was found.