The value of ∑200k = 99k3∑k = 99200k3 = _____.

Multiple Choice

380,999,777

380,977,779

380,477,799

380,477,779

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

The value of ∑200k = 99k^3∑k = 99200k^3 = _____.
Multiple Choice: 380,999,777, 380,977,779, 380,477,799, 380,477,779

Explore the detailed solution for calculating the sum of cubes from k=1 to 200 using the formula ∑k^3 = (n(n+1)/2)^2. This math problem is ideal for high school students (Grades 10-12) and covers key concepts like summation and series.

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