Find the odd term out: 3, 7, 15, 39, 63, 127, 255, 511 Exactly one number breaks the intended “2^n − 1” pattern.

Introduction / Context:
Many fast-growing series follow the form 2^n − 1: 3, 7, 15, 31, 63, 127, 255, 511, … Spotting any deviation from this canonical list identifies the odd man out.

Given Data / Assumptions:

• Presented sequence: 3, 7, 15, 39, 63, 127, 255, 511.
• Known reference pattern: 3, 7, 15, 31, 63, 127, 255, 511.
• Exactly one number is inconsistent.

Concept / Approach:
Compare term-by-term with 2^n − 1 for n = 2 through 9. The mismatch pinpoints the erroneous entry.

Step-by-Step Solution:
2^2 − 1 = 3 ✓2^3 − 1 = 7 ✓2^4 − 1 = 15 ✓2^5 − 1 = 31 (but the series shows 39 ✗)2^6 − 1 = 63 ✓2^7 − 1 = 127 ✓2^8 − 1 = 255 ✓2^9 − 1 = 511 ✓

Verification / Alternative check:
Replacing 39 with 31 yields a perfect 2^n − 1 run. No other power-based family fits all remaining values so cleanly.

Why Other Options Are Wrong:

• 7 / 15 / 63 belong exactly to 2^3 − 1, 2^4 − 1, 2^6 − 1 respectively and are consistent.

Common Pitfalls:

• Assuming a multiplicative rule like “×2 + 1” inconsistently; the strict 2^n − 1 check is definitive.

Final Answer:
39