Number series: 3, 5, 9, 17, 33, 65, ? Identify the next term by analyzing the pattern of successive differences.

Introduction / Context:
Many number-series questions are built on differences or ratios that themselves follow a pattern. Here the terms grow moderately fast, suggesting either multiplicative growth or additive jumps that expand regularly.

Given Data / Assumptions:

• Series: 3, 5, 9, 17, 33, 65, ?
• We assume a single-rule deterministic pattern applies to all steps.

Concept / Approach:
Compute successive differences and then look for a clear, simple law (e.g., constant, arithmetic progression, geometric progression, powers of 2). If the first differences themselves form a recognizable series, extend that series one step to project the next main term.

Step-by-Step Solution:
Differences: 5−3 = 2, 9−5 = 4, 17−9 = 8, 33−17 = 16, 65−33 = 32.These differences are 2, 4, 8, 16, 32, i.e., powers of 2 doubling each time.Next difference = 64.Next term = 65 + 64 = 129.

Verification / Alternative check:
Project backwards: starting at 3, repeatedly add 2^n (n = 1, 2, 3, 4, 5, 6). This rebuilds the sequence through 65 and yields 129 next, confirming consistency.

Why Other Options Are Wrong:
128, 126, and 132 ignore the doubling-difference structure. 128 would require adding 63, which breaks the powers-of-two pattern.

Common Pitfalls:
Trying to fit a direct multiplicative pattern (e.g., times something plus/minus) when a simple difference pattern exists; stopping after checking only one or two gaps.

Final Answer:
129