`(11^3 + 12^3 + 13^3 + ...... + 30^3) = ?`

Options

A. 231200
B. 223100
C. 213020
D. 213200

Answer

To solve the problem of finding the sum of cubes from \(11^3\) to \(30^3\), we can use the formula for the sum of cubes from \(1^3\) to \(n^3\), which is given by:

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

### Step-by-Step Solution:

1. **Calculate the sum of cubes from \(1^3\) to \(30^3\)**:
- Here, \(n = 30\).
- Using the formula:
\[ \text{Sum}_{1 \text{ to } 30} = \left(\frac{30 \times 31}{2}\right)^2 \]
- Calculate \(30 \times 31 = 930\).
- Now, calculate \(\frac{930}{2} = 465\).
- Finally, calculate \(465^2\):
\[ 465^2 = 216225 \]

2. **Calculate the sum of cubes from \(1^3\) to \(10^3\)**:
- Here, \(n = 10\).
- Using the formula:
\[ \text{Sum}_{1 \text{ to } 10} = \left(\frac{10 \times 11}{2}\right)^2 \]
- Calculate \(10 \times 11 = 110\).
- Now, calculate \(\frac{110}{2} = 55\).
- Finally, calculate \(55^2\):
\[ 55^2 = 3025 \]

3. **Subtract the sum of cubes from \(1^3\) to \(10^3\) from the sum of cubes from \(1^3\) to \(30^3\)**:
\[ \text{Sum}_{11 \text{ to } 30} = \text{Sum}_{1 \text{ to } 30} - \text{Sum}_{1 \text{ to } 10} \]
\[ \text{Sum}_{11 \text{ to } 30} = 216225 - 3025 \]
- Perform the subtraction:
\[ 216225 - 3025 = 213200 \]

Thus, the sum of cubes from \(11^3\) to \(30^3\) is \(213200\).

### Final Answer:
\[ (11^3 + 12^3 + 13^3 + \ldots + 30^3) = 213200 \]