If `1^(3)+2^(3)+…. + 10^(3)=3025`, then the value of `2^(3)+4^(3)+…. + 20^(3)` is :

Options

A. `7590`
B. `5060`
C. `24200`
D. `12100`

Answer

To solve the problem, we need to find the value of \(2^3 + 4^3 + 6^3 + 8^3 + 10^3 + 12^3 + 14^3 + 16^3 + 18^3 + 20^3\).

We know from the problem statement that:

\[ 1^3 + 2^3 + 3^3 + \ldots + 10^3 = 3025 \]

Now, we can express \(2^3 + 4^3 + 6^3 + 8^3 + 10^3 + 12^3 + 14^3 + 16^3 + 18^3 + 20^3\) in terms of the sum of cubes of the first 10 natural numbers.

### Step 1: Rewrite the sum of cubes
Notice that:
\[ 2^3 + 4^3 + 6^3 + 8^3 + 10^3 + 12^3 + 14^3 + 16^3 + 18^3 + 20^3 = 2^3(1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3) \]

### Step 2: Factor out \(2^3\)
This can be factored as:
\[ = 2^3 \cdot (1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3) \]
Since \(2^3 = 8\), we have:
\[ = 8 \cdot (1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3) \]

### Step 3: Substitute the value of the sum of cubes
Now, substituting the known value:
\[ = 8 \cdot 3025 \]

### Step 4: Calculate the final result
Now we calculate:
\[ 8 \cdot 3025 = 24200 \]

Thus, the value of \(2^3 + 4^3 + 6^3 + 8^3 + 10^3 + 12^3 + 14^3 + 16^3 + 18^3 + 20^3\) is \(24200\).

### Final Answer
\[ \text{The value of } 2^3 + 4^3 + 6^3 + 8^3 + 10^3 + 12^3 + 14^3 + 16^3 + 18^3 + 20^3 = 24200 \]