The sum `(5^3+6^3+……..10^3)` is equal to

Options

A. 2295
B. 2425
C. 2495
D. 2925

Answer

To find the sum \(5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3\), we can use the formula for the sum of cubes of the first \(n\) natural numbers, which is given by:

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

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

We first calculate the sum of cubes from \(1\) to \(10\):

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

Calculating this:

\[ = \left( \frac{10 \times 11}{2} \right)^2 = (55)^2 = 3025 \]

### Step 2: Calculate the sum of cubes from \(1^3\) to \(4^3\)

Next, we calculate the sum of cubes from \(1\) to \(4\):

\[ \text{Sum from } 1^3 \text{ to } 4^3 = \left( \frac{4(4+1)}{2} \right)^2 \]

Calculating this:

\[ = \left( \frac{4 \times 5}{2} \right)^2 = (10)^2 = 100 \]

### Step 3: Subtract the sum of cubes from \(1^3\) to \(4^3\) from the sum of cubes from \(1^3\) to \(10^3\)

Now, we subtract the sum of cubes from \(1^3\) to \(4^3\) from the sum of cubes from \(1^3\) to \(10^3\):

\[ 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3 = 3025 - 100 = 2925 \]

### Final Answer

Thus, the sum \(5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3\) is equal to:

\[ \boxed{2925} \]