What comes next: 2, 3, 5, 9, 17, 33, ?

Correct — A.
Rule: double the term, then subtract 1 (× 2 − 1).

2 × 2 − 1 = 3, and 3 × 2 − 1 = 5
5 × 2 − 1 = 9, and 9 × 2 − 1 = 17
17 × 2 − 1 = 33

Next: 33 × 2 − 1 = 65 → option (a).

Why the others are wrong

• (b)
  63 — 63 needs a last difference of 30. The differences run 1, 2, 4, 8, 16 and double each time, so the next is 32, and 33 + 32 = 65.
• (c)
  61 — 61 adds 28 to 33. With the differences doubling to 32, 61 falls 4 short of 65.
• (d)
  67 — 67 is 33 × 2 + 1: the sign of the correction is flipped. Every given step uses − 1, as in 17 × 2 − 1 = 33.

When a series grows by roughly doubling, test × 2 with a small correction. Here every term is twice the one before, minus 1.

The differences tell the same story: 1, 2, 4, 8, 16, each twice the last, so the next difference is 32. A third view: every term is a power of 2 plus 1, from 2⁰ + 1 = 2 up to 2⁶ + 1 = 65.

Key facts

• Here each term is 2 × (previous term) − 1.
• The differences 1, 2, 4, 8, 16 double, so the next difference is 32.
• The terms are 2ⁿ + 1 for n = 0 to 6, ending at 65.

Common traps

• Flipping the correction and computing 33 × 2 + 1 = 67
• Adding 30 instead of doubling the last difference to 32, which gives 63

No directly related past PYQ was found.