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

Options

A. an even integer
B. 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. The sum of cubes from \( 1^3 \) to \( n^3 \) is given by:

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

### Step 1: Calculate the sum of cubes from \( 1^3 \) to \( 20^3 \)

Using the formula:

\[ \text{Sum from } 1^3 \text{ to } 20^3 = \left( \frac{20 \times 21}{2} \right)^2 \]

Calculating the value:

\[ = \left( \frac{420}{2} \right)^2 = (210)^2 = 44100 \]

### Step 2: Calculate the sum of cubes from \( 1^3 \) to \( 10^3 \)

Using the same formula:

\[ \text{Sum from } 1^3 \text{ to } 10^3 = \left( \frac{10 \times 11}{2} \right)^2 \]

Calculating the value:

\[ = \left( \frac{110}{2} \right)^2 = (55)^2 = 3025 \]

### Step 3: Subtract the sum from \( 1^3 \) to \( 10^3 \) from the sum from \( 1^3 \) to \( 20^3 \)

Now we can find the sum from \( 11^3 \) to \( 20^3 \):

\[ 11^3 + 12^3 + 13^3 + \ldots + 20^3 = \text{Sum from } 1^3 \text{ to } 20^3 - \text{Sum from } 1^3 \text{ to } 10^3 \]

Calculating the final value:

\[ = 44100 - 3025 = 41075 \]

### Final Answer

Thus, the value of \( 11^3 + 12^3 + 13^3 + \ldots + 20^3 \) is \( \boxed{41075} \).

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