What is the value of `(6^3+7^3+8^3+…….+10^3)` ?

Options

A. 3025
B. 2800
C. 2900
D. 2925

Answer

To find the value of \(6^3 + 7^3 + 8^3 + 9^3 + 10^3\), we can use the formula for the sum of cubes of the first \(n\) natural numbers. The formula is given by:

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

### Step-by-step Solution:

1. **Identify the range of cubes**: We need to calculate the sum of cubes from \(6\) to \(10\). This can be expressed as:
\[ 6^3 + 7^3 + 8^3 + 9^3 + 10^3 \]

2. **Use the formula for sum of cubes**: The sum of cubes from \(1\) to \(n\) can be calculated using the formula:
\[ S(n) = \left( \frac{n(n+1)}{2} \right)^2 \]
We will calculate \(S(10)\) and \(S(5)\) and then find the difference to get \(S(6 \text{ to } 10)\).

3. **Calculate \(S(10)\)**:
\[ S(10) = \left( \frac{10 \times 11}{2} \right)^2 = \left( \frac{110}{2} \right)^2 = 55^2 = 3025 \]

4. **Calculate \(S(5)\)**:
\[ S(5) = \left( \frac{5 \times 6}{2} \right)^2 = \left( \frac{30}{2} \right)^2 = 15^2 = 225 \]

5. **Calculate the sum from \(6\) to \(10\)**:
\[ S(6 \text{ to } 10) = S(10) - S(5) = 3025 - 225 = 2800 \]

### Final Answer:
Thus, the value of \(6^3 + 7^3 + 8^3 + 9^3 + 10^3\) is \(2800\).