What is the value of ` 1^(3) + 2^(3) + 3^(3) + . . . . + 10^(3)` ?

Options

A. 5500
B. 3025
C. 6025
D. 2975

Answer

To find the value of \( 1^3 + 2^3 + 3^3 + \ldots + 10^3 \), we can use the formula for the sum of cubes of the first \( n \) natural numbers, which is given by:

\[ \left( \frac{n(n + 1)}{2} \right)^2 \]

### Step-by-Step Solution:

1. **Identify \( n \)**: In this case, we need to find the sum up to \( 10^3 \), so \( n = 10 \).

2. **Substitute \( n \) into the formula**:
\[ \text{Sum} = \left( \frac{10(10 + 1)}{2} \right)^2 \]

3. **Calculate \( 10 + 1 \)**:
\[ 10 + 1 = 11 \]

4. **Calculate \( 10 \times 11 \)**:
\[ 10 \times 11 = 110 \]

5. **Divide \( 110 \) by \( 2 \)**:
\[ \frac{110}{2} = 55 \]

6. **Square the result**:
\[ 55^2 = 3025 \]

Thus, the value of \( 1^3 + 2^3 + 3^3 + \ldots + 10^3 \) is \( 3025 \).

### Final Answer:
\[ \text{The value is } 3025. \]