Find the incorrect term in the following number series. Series: 3, 7, 15, 39, 63, 127, 255, 511

Introduction / Context: A classic series uses Mersenne numbers of the form 2^n − 1. If most terms match 2^n − 1 and one does not, that term is the wrong entry.

Given Data / Assumptions:

• Series: 3, 7, 15, 39, 63, 127, 255, 511
• Target pattern: 2^2 − 1 = 3, 2^3 − 1 = 7, 2^4 − 1 = 15, 2^5 − 1 = 31, 2^6 − 1 = 63, …

Concept / Approach: Compare each term with 2^n − 1 for consecutive n. Any deviation from this exact form identifies the misfit.

Step-by-Step Solution: 3 = 2^2 − 1; 7 = 2^3 − 1; 15 = 2^4 − 1. Expected next is 31 = 2^5 − 1, but the series lists 39 (incorrect). Continuing: 63 = 2^6 − 1; 127 = 2^7 − 1; 255 = 2^8 − 1; 511 = 2^9 − 1.

Verification / Alternative check: Replacing 39 with 31 yields a perfect run of Mersenne numbers from n = 2 to 9.

Why Other Options Are Wrong: 15, 63, 127 all exactly match 2^n − 1 for n = 4, 6, 7 respectively; they are correct members.

Common Pitfalls: Looking for ad-hoc multipliers instead of recognizing the well-known 2^n − 1 pattern.

Final Answer: 39