Find the incorrect term in the following number series. Series: 5, 27, 61, 122, 213, 340, 509

Introduction / Context: Many crafted series follow a cubic template like n^3 − 3. If most terms fit a simple formula and one does not, that outlier is the incorrect term.

Given Data / Assumptions:

• Series: 5, 27, 61, 122, 213, 340, 509
• We test terms against the model a(n) = k^3 − 3 for consecutive integers k.

Concept / Approach: Check each term against k^3 − 3 starting with k = 4 (since 4^3 − 3 = 64 − 3 = 61, which appears). If all others match, the remaining non-matching term is wrong.

Step-by-Step Solution: 61 = 4^3 − 3 122 = 5^3 − 3 213 = 6^3 − 3 340 = 7^3 − 3 509 = 8^3 − 3 But 27 = 3^3, not 3^3 − 3 (which would be 24). Therefore 27 breaks the rule.

Verification / Alternative check: Replacing 27 by 24 yields a perfect run of k^3 − 3 for k = 3 to 8: 24, 61, 122, 213, 340, 509.

Why Other Options Are Wrong: 61, 122, 509 all exactly equal k^3 − 3 for k = 4, 5, 8 respectively.

Common Pitfalls: Assuming a single difference pattern rather than testing a direct closed-form formula.

Final Answer: 27