What will come in the place of question (?) mark in the following number…

Concept: In a difference-based number series, first list the gaps between consecutive terms and look for a rule that generates each gap from the one before it. A common rule is a linear recurrence on the differences, d(n) = 2 × d(n-1) + 1, where every gap is one more than double the previous gap.

Application: Start from the gaps that can be computed directly in 80, ?, 117, 168, 271, 478, then run the rule backwards to recover the missing gaps.

1. 168 − 117 = 51, 271 − 168 = 103, and 478 − 271 = 207.

2. Confirm the rule on these known gaps: 2 × 51 + 1 = 103 and 2 × 103 + 1 = 207, so d(n) = 2 × d(n-1) + 1 holds.

3. Recover the gap just before 51 by solving 2 × d + 1 = 51, which gives d = 25; this is the gap from the missing term up to 117, so the missing term is 117 − 25 = 92.

4. Recover one more gap by solving 2 × d + 1 = 25, which gives d = 12, the gap from 80 to the missing term, so 80 + 12 = 92 as well. Both directions agree.

Cross-check: With 92 placed, the complete difference list is 12, 25, 51, 103, 207, an unbroken chain where each value equals twice the previous plus 1. As a second test, every term equals double the previous term minus a constant that falls steadily as 68, 67, 66, 65, 64, which independently confirms the result.