`11^(3)+12^(3)+13^(3)+`…………………. `+20^(3)` is

Options

A. an even integer
B. an odd integer divisible by 5
C. multiple of 10
D. odd integer but not a multiple of 5.

Answer

To solve the problem \( 11^3 + 12^3 + 13^3 + \ldots + 20^3 \), we can use the formula for the sum of cubes and adjust it for the specific range we need.

### Step-by-Step Solution:

1. **Identify the Range**: We need to find the sum of cubes from \( 11 \) to \( 20 \).

2. **Use the Sum of Cubes Formula**: The sum of the first \( n \) cubes is given by:
\[ S_n = \left( \frac{n(n+1)}{2} \right)^2 \]
This means we can calculate the sum of cubes from \( 1^3 \) to \( 20^3 \) and subtract the sum from \( 1^3 \) to \( 10^3 \).

3. **Calculate \( S_{20} \)**:
\[ S_{20} = \left( \frac{20 \times 21}{2} \right)^2 = (210)^2 = 44100 \]

4. **Calculate \( S_{10} \)**:
\[ S_{10} = \left( \frac{10 \times 11}{2} \right)^2 = (55)^2 = 3025 \]

5. **Subtract \( S_{10} \) from \( S_{20} \)**:
\[ S_{11 \text{ to } 20} = S_{20} - S_{10} = 44100 - 3025 = 41075 \]

6. **Final Result**:
\[ 11^3 + 12^3 + 13^3 + \ldots + 20^3 = 41075 \]