Understanding the Sum of Cubes Problem

This question involves the relationship between the sum of the first 'n' natural numbers and the sum of their cubes. We are given the sum of the cubes and asked to find the square of the sum of the numbers themselves.

Key Mathematical Formulas

Let's recall the standard formulas for the sum of the first 'n' natural numbers and the sum of the cubes of the first 'n' natural numbers:

• The sum of the first 'n' natural numbers ($S_n$) is given by: $ S_n = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} $
• The sum of the cubes of the first 'n' natural numbers ($C_n$) is given by: $ C_n = 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2 $

Notice that the formula for the sum of cubes ($C_n$) is the square of the formula for the sum of the first 'n' natural numbers ($S_n$). This means:

Step-by-Step Solution

1. Identify Given Information: The problem states that the sum of the cubes of the first 'n' natural numbers is 784. So, we have: $ C_n = 784 $
2. Identify What Needs to be Found: We need to find the square of the sum of those 'n' numbers. This is represented as $(S_n)^2$.
3. Apply the Relationship: Using the relationship $C_n = (S_n)^2$, we can substitute the given value of $C_n$: $ (S_n)^2 = C_n = 784 $
4. Determine the Result: The question asks directly for the value of the square of the sum of the first 'n' natural numbers, which is $(S_n)^2$. From the previous step, we found that $(S_n)^2 = 784$.

Conclusion

Therefore, the square of the sum of the first 'n' natural numbers is exactly the sum of their cubes. Since the sum of the cubes is given as 784, the square of the sum of the numbers is also 784.