What should come in place of the question mark (?) in the given series? 9 17 33 65 129 ?

Correct — B. Rule: double the term and subtract 1. Equivalently the gaps are 8, 16, 32 and 64, doubling each step.

9 × 2 - 1 = 17
17 × 2 - 1 = 33
33 × 2 - 1 = 65
65 × 2 - 1 = 129

129 × 2 - 1 = 257, and the gap reading agrees, since 129 + 128 = 257. That is option (b).

Why the others are wrong

• (a)
  259 — 259 needs a step of 130 from 129. The gaps so far are 8, 16, 32 and 64 — powers of two — so the next gap is 128.
• (c)
  258 — 258 is 129 doubled with the subtraction dropped. That minus one is what keeps every term odd, and 258 is even.
• (d)
  254 — 254 implies a gap of 125, which neither doubles 64 nor fits anything earlier. It is also even, and every term in this series is odd.

When a series roughly doubles, take the differences first. Here they run 8, 16, 32, 64, and a difference series that doubles points to a term rule of the form 2n + c.

Fix c from any single step: 9 doubled is 18 and the next term is 17, so c is -1.

The rule 2n - 1 also explains why every term is odd and why each term is one more than a power of two: 9 = 8 + 1, 17 = 16 + 1, 33 = 32 + 1.

Key facts

• Each term is one more than a power of two, running 8 + 1, 16 + 1, 32 + 1, 64 + 1, 128 + 1.
• The rule 2n - 1 turns 129 into 257.
• Every term of the series is odd, which rules out 258 and 254 at sight.

Common traps

• Doubling the last term and forgetting the minus one, which gives 258.
• Adding 130 because the gap is estimated from 129 rather than from 64.
• Reading the gaps as 8, 16, 32, 64 and then adding 64 again.

No directly related past PYQ was found.