Question

For all \( n \in \mathbb{N} \), if \( 1^3 + 2^3 + 3^3 + \cdots + n^3>x \), then a value of \( x \) among the following is:

• A. \(\frac{n^2}{4}\)
• B. \(n^2\)
• C. \(n^4\)
• D. \(\frac{n^2 (n+1)^2}{4}\)

Hint:

Recall the formula for sum of cubes: \(\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2\).

Answer 1

The Correct Option is A

Step 1: Use sum of cubes formula
\[ 1^3 + 2^3 + 3^3 + \cdots + n^3 = \left( \frac{n(n+1)}{2} \right)^2 = \frac{n^2 (n+1)^2}{4} \] Step 2: Inequality given
\[ \left( \frac{n(n+1)}{2} \right)^2>x \] Step 3: Choose a value \( x \) less than this sum
Among the options, \(\frac{n^2}{4}\) is smaller than the sum and satisfies the condition.