Cube-step analogy 8 → 28 corresponds to a cube-step + 1 pattern. Find the analogous value for 27 → ?

Introduction / Context:
This analogy deals with perfect cubes and a predictable “next-cube plus one” transformation. Recognizing cube numbers is the essential first step.

Given Data / Assumptions:

• 8 is 2^3, and 27 is 3^3.
• The mapping for 8 is to 28.

Concept / Approach:
If 8 (which is 2^3) maps to 28, notice that 28 = 27 + 1 = 3^3 + 1. Therefore, the rule appears to be: k^3 maps to (k+1)^3 + 1.

Step-by-Step Solution:
Start with 27 = 3^3.Next cube is 4^3 = 64.Add 1 as per the pattern → 64 + 1 = 65.

Verification / Alternative check:
Reapply the rule to 8: 8 = 2^3, next cube is 3^3 = 27, add 1 → 28. This confirms the transformation.

Why Other Options Are Wrong:
64: Misses the final “+1”.28 and 8: Mix values from the example rather than applying the rule to 27.

Common Pitfalls:
Identifying 27 as 9^3 instead of 3^3 or forgetting to add 1 after taking the next cube.

Final Answer:
65