To solve this, we need to calculate the value of the sum $\sum_{k=1}^{200} k^3$.

Step 1: Formula for the sum of cubes

The sum of cubes of the first $n$ natural numbers is given by the formula:

$\sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2$

For $n = 200$, we can substitute into the formula:

$\sum_{k=1}^{200} k^3 = \left( \frac{200(200+1)}{2} \right)^2 = \left( \frac{200 \times 201}{2} \right)^2$

$= (100 \times 201)^2 = 20100^2$

Step 2: Compute $20100^2$

Now, let's calculate:

$20100^2 = 404010000$

Step 3: Adjust according to the equation

The problem provides $\sum_{k=1}^{200} k^3 = 99 \times 200k^3$, which implies multiplying this sum by 99:

$99 \times 404010000 = 39996990000$

However, this leads to a mismatch in context. Please double check for clarity or different forms for