This mathematical equation represents a summation of products of consecutive integers, generalized into a formula. The left-hand side sums up terms of the form $k(k+1)(k+2)$, where $k$ ranges from $1$ to $n$. The right-hand side provides a closed-form expression for the summation.

The summation: $\sum_{k=1}^n k(k+1)(k+2)$ is simplified and expressed as: $\frac{n(n+1)(n+2)(n+3)}{4}.$

Explanation:

1. Structure of the summation: Each term $k(k+1)(k+2)$ is a product of three consecutive integers.
2. Closed-form derivation: This result is derived using properties of summations and polynomials. Specific formulas for sums of integers ($\sum k$), squares ($\sum k^2$), and cubes ($\sum k^3$) are used in this derivation.

Would you like me to derive this result in detail, prove it step by step, or explain the individual summation components?

Questions to Explore:

1. How do summations like $\sum k^p$ (e.g., $p = 1, 2, 3$) influence the derivation?
2. Can this formula be generalized for $k(k+1)(k+2)\dots(k+r)$ for $r > 2$?
3. Why does the closed form involve a division by 4?
4. How can mathematical induction verify this closed form?
5. What are practical applications of such summations?

Tip:

To simplify complex summations, break them into smaller components and use known formulas like $\sum k^p$. This approach saves time and avoids errors.