Choose the correct alternative that will complete the given number series. 9,…

Concept: many number series are generated by a first-order recurrence — each term is produced from the one before it via a fixed multiplier combined with a changing additive term that itself follows a simple pattern, most commonly the sequence of consecutive odd numbers (1, 3, 5, 7, 9, ...). Solving such a series means first confirming this rule holds across every consecutive pair of given terms, not just the first two, and only then applying it forward to the missing term, with an independent check against any term given after the gap.

Applying the rule to 9, 17, 31, 57, ?, 205:

1. 9 × 2 − 1 = 18 − 1 = 17

2. 17 × 2 − 3 = 34 − 3 = 31

3. 31 × 2 − 5 = 62 − 5 = 57

4. 57 × 2 − 7 = 114 − 7 = 107, the missing term

Cross-check: applying the same rule one more step from the computed value must reproduce the series' given sixth term.

• 107 × 2 − 9 = 214 − 9 = 205, exactly matching the given sixth term.

• Independent second-difference check: the term-to-term differences are 8, 14, 26 — their own differences double each time (6, then 12), so the next difference is 26 + 24 = 50, giving 57 + 50 = 107, the same result.

Both the forward recurrence and the second-difference check confirm the missing term is 107.