write in the closed form (evaluate) : sum(k=1)^n k(k^2+2)

Let's evaluate the sum \( S = \sum_{k=1}^n k(k^2 + 2) \).

First, expand the expression inside the summation:

\[
S = \sum_{k=1}^n k(k^2 + 2) = \sum_{k=1}^n (k^3 + 2k)
\]

Now, we can split this into two separate sums:

\[
S = \sum_{k=1}^n k^3 + 2\sum_{k=1}^n k
\]

### 1. Sum of \( k^3 \):

The sum of the cubes of the first \( n \) natural numbers is given by:

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

### 2. Sum of \( k \):

The sum of the first \( n \) natural numbers is given by:

\[
\sum_{k=1}^n k = \frac{n(n+1)}{2}
\]

### Combine the results:

Now substitute these into the original sum:

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

Simplify the second term:

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

Finally, you can leave the expression in this form or expand it further if needed:

\[
S = \frac{n^2(n+1)^2}{4} + n(n+1)
\]

Alternatively, you can factor the expression further:

\[
S = \frac{n(n+1)}{4} \left[ n(n+1) + 4 \right]
\]

Would you like a detailed explanation of each step or have any questions?

Here are 5 related questions to explore further:
1. How would you calculate the sum \( \sum_{k=1}^n k^2(k^2+3) \)?
2. What is the sum \( \sum_{k=1}^n (k^4 + 3k^2) \) in closed form?
3. Can you derive the formula for \( \sum_{k=1}^n k^4 \) from scratch?
4. How does the sum change if you have \( k(k^2 + 3) \) instead of \( k(k^2 + 2) \)?
5. What is the general form of \( \sum_{k=1}^n k(k^2 + a) \) for any constant \( a \)?

**Tip:** When dealing with summations, breaking them into simpler sums and using known formulas often simplifies the problem significantly.

Learn how to evaluate the sum \( S = \sum_{k=1}^n k(k^2 + 2) \) with a detailed step-by-step solution. This problem involves calculating the sum of cubes and arithmetic series. Suitable for advanced high school students and above.

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