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.