We are tasked with proving the following statement using Principle of Mathematical Induction (PMI):

$S(n): 1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + n(n + 1)(n + 2) = \frac{n(n + 1)(n + 2)(n + 3)}{4}$

Step 1: Base Case (n = 1)

For $n = 1$, we have:

$S(1): 1 \cdot 2 \cdot 3 = \frac{1(1 + 1)(1 + 2)(1 + 3)}{4}$

Simplifying both sides:

$1 \cdot 2 \cdot 3 = 6$

And on the right-hand side:

$\frac{1 \cdot 2 \cdot 3 \cdot 4}{4} = \frac{24}{4} = 6$

Thus, $S(1)$ holds true.

Step 2: Inductive Hypothesis

Assume that the statement is true for $n = k$. That is, we assume:

$S(k): 1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + k(k + 1)(k + 2) = \frac{k(k + 1)(k + 2)(k + 3)}{4}$

Step 3: Inductive Step

We need to prove that if $S(k)$ is true, then $S(k + 1)$ is also true. That is, we need to show:

$1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3) = \frac{(k + 1)(k + 2)(k + 3)(k + 4)}{4}$

Using the inductive hypothesis, we can substitute for the sum up to $k$:

$\frac{k(k + 1)(k + 2)(k + 3)}{4} + (k + 1)(k + 2)(k + 3)$

Factor out $(k + 1)(k + 2)(k + 3)$ from the expression:

$(k + 1)(k + 2)(k + 3) \left( \frac{k}{4} + 1 \right)$

Simplify the expression inside the parentheses:

$\frac{k}{4} + 1 = \frac{k + 4}{4}$

So we have:

$(k + 1)(k + 2)(k + 3) \cdot \frac{k + 4}{4}$

This simplifies to:

$\frac{(k + 1)(k + 2)(k + 3)(k + 4)}{4}$

Thus, $S(k + 1)$ holds true, completing the inductive step.

Step 4: Conclusion

By the principle of mathematical induction, we have shown that for every positive integer $n$, the statement:

$1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + n(n + 1)(n + 2) = \frac{n(n + 1)(n + 2)(n + 3)}{4}$

is true.

---

Would you like a deeper explanation of any step or have any other questions?

Here are 5 related questions to think about:

1. How does mathematical induction help in proving statements about sequences?
2. Can you use PMI to prove other polynomial identities?
3. How would the result change if the summation involved different powers of $n$?
4. How does factoring simplify the inductive step in PMI proofs?
5. What is the difference between strong and weak mathematical induction?

Tip: Always make sure to check the base case carefully in PMI, as it anchors the whole proof.