What is the average of the cubes of the first five natural numbers?

Introduction:
You are asked to compute the mean value of 1^3, 2^3, 3^3, 4^3, and 5^3. This can be done either by direct calculation and averaging or by recalling the identity for the sum of cubes. We will use explicit values to keep it transparent.

Given Data / Assumptions:

• First five natural numbers: 1, 2, 3, 4, 5.
• Cubes: 1, 8, 27, 64, 125.

Concept / Approach:
Average = (sum of values) / (count). Compute the five cubes, add them, and divide by 5. This is straightforward and avoids memorizing formulas.

Step-by-Step Solution:

Verification / Alternative check:
Using the identity (1 + 2 + 3 + 4 + 5)^2 = (sum of cubes) for consecutive starting at 1, we have (15)^2 = 225, confirming the sum before division by 5.

Why Other Options Are Wrong:
55, 65, 35, and 50 do not match the computed mean from the explicit cube values.

Common Pitfalls:
Confusing squares and cubes, or taking an unweighted average by mistake. Ensure all five cubes are included.

Final Answer:
45