In the following series, identify the wrong term: 3, 7, 15, 39, 63, 127, 255, 511.

Introduction / Context:
This question is about finding the one wrong term in what is almost a standard number pattern. Many exam series are based on well known sequences such as powers of two. You are expected to recognize the structure and then spot the element that does not fit.

Given Data / Assumptions:

• Series given: 3, 7, 15, 39, 63, 127, 255, 511.
• Exactly one term is incorrect.
• The pattern is expected to be simple, often exponential or near exponential.

Concept / Approach:
Looking at the terms, they are close to powers of two minus one. For example, 3 is 2^2 - 1, 7 is 2^3 - 1, 15 is 2^4 - 1, and 31 would be 2^5 - 1. This strongly suggests that the intended pattern is 2^n - 1 for consecutive n values. Any deviation from this identity shows the wrong term.

Step-by-Step Solution:
Step 1: Express each term as 2^n - 1 where possible. 3 = 2^2 - 1. 7 = 2^3 - 1. 15 = 2^4 - 1. The next expected term for n = 5 is 2^5 - 1 = 32 - 1 = 31. Step 2: Compare with the given fourth term, which is 39 instead of 31. Step 3: Continue to verify remaining terms. 63 = 2^6 - 1. 127 = 2^7 - 1. 255 = 2^8 - 1. 511 = 2^9 - 1. Step 4: All other terms match the pattern 2^n - 1 perfectly, except 39.

Verification / Alternative check:
Write down the ideal sequence using 2^n - 1 for n from 2 to 9: 3, 7, 15, 31, 63, 127, 255, 511. Compare this with the given sequence: 3, 7, 15, 39, 63, 127, 255, 511. The only discrepancy is the fourth position, confirming that the wrong term is 39. No other term needs to be altered for the pattern to hold from start to finish.

Why Other Options Are Wrong:
63, 127, 255, and 511 are all exactly one less than powers of two. For instance, 63 is 64 - 1, 127 is 128 - 1, 255 is 256 - 1, and 511 is 512 - 1. If any of these were changed, the power of two minus one structure would be broken. Therefore none of these values can be considered incorrect in the context of this pattern.

Common Pitfalls:
Some learners try to form complicated differences or ratios and may mistakenly conclude that a later term is wrong. The safer method is to notice that most values already fit a well known pattern. Once that pattern is recognized, the outlier becomes obvious. Always check for common sequences like powers of two, squares, or factorial adjustments before inventing a complex rule.

Final Answer:
The only number that does not follow the 2^n - 1 pattern is 39, so 39 is the wrong term in the series.