In the number series 7, 28, 63, 124, 215, 342, 511, exactly one term is wrong. Determine which number does not fit the underlying pattern.

Introduction / Context:
This question presents a series of moderately large numbers: 7, 28, 63, 124, 215, 342, 511. We are told that exactly one term is incorrect. Many such series are built from cubic expressions such as n^3 - 1 or n^3 + 1. Our task is to compare these numbers with a likely cubic pattern and identify the outlier.

Given Data / Assumptions:

• Series: 7, 28, 63, 124, 215, 342, 511.
• Exactly one number is incorrect.
• Cubic patterns are common in such tasks, especially of the form n^3 - 1.

Concept / Approach:
We test whether each term can be expressed as n^3 - 1 for consecutive integer values of n. If almost every term matches that form except one, that unmatched number is the wrong term. This approach is simple but very powerful for sequences that resemble slightly less than perfect cubes.

Step-by-Step Solution:
1. Compute n^3 - 1 for consecutive n values and compare with the series. For n = 2: 2^3 - 1 = 8 - 1 = 7. For n = 3: 3^3 - 1 = 27 - 1 = 26. For n = 4: 4^3 - 1 = 64 - 1 = 63. For n = 5: 5^3 - 1 = 125 - 1 = 124. For n = 6: 6^3 - 1 = 216 - 1 = 215. For n = 7: 7^3 - 1 = 343 - 1 = 342. For n = 8: 8^3 - 1 = 512 - 1 = 511. 2. So the ideal n^3 - 1 sequence from n = 2 to 8 is: 7, 26, 63, 124, 215, 342, 511. 3. Compare with the given sequence: 7, 28, 63, 124, 215, 342, 511. 4. All terms match except the second one: it should be 26, but the series has 28. 5. Therefore 28 is the wrong term.

Verification / Alternative check:
Write the corrected series explicitly: 7, 26, 63, 124, 215, 342, 511. Check: 7 = 2^3 - 1, 26 = 3^3 - 1, 63 = 4^3 - 1, 124 = 5^3 - 1, 215 = 6^3 - 1, 342 = 7^3 - 1, 511 = 8^3 - 1. Every corrected term now fits the same simple cubic pattern perfectly.

Why Other Options Are Wrong:

• 124, 215, 342: Each of these matches n^3 - 1 for some integer n, so they are consistent with the series structure.
• For example, 124 equals 5^3 - 1 and 342 equals 7^3 - 1, so they cannot be the wrong term.

Common Pitfalls:
Some candidates attempt to find patterns in the differences between terms, but for cubic based sequences the differences themselves are not simple enough to spot easily. This can waste time and lead to wrong guesses. Instead, whenever numbers lie near perfect cubes, always test expressions like n^3 ± 1 or n^3 ± 2 first.

Final Answer:
The wrong term in the series is 28.