This is a mathematical induction problem written in Ukrainian. The task is to prove, using the method of mathematical induction, that for any natural number $n$, the following equation holds:

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

Steps for Mathematical Induction:

We will prove the statement for all $n \geq 1$ by following two steps:

1. Base case: Prove the formula holds for $n = 1$.
2. Inductive step: Assume the formula holds for $n = k$ and prove that it holds for $n = k+1$.

Base Case $n = 1$:

For $n = 1$, the left-hand side of the equation is:

$1 \cdot 2 \cdot 3 = 6$

The right-hand side of the equation is:

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

Thus, the formula holds for $n = 1$.

Inductive Step:

Assume the formula holds for $n = k$, i.e.,

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

We need to prove that the formula holds for $n = k + 1$, i.e.,

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

Starting with the left-hand side, use the induction hypothesis:

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

Factor out $(k+1)(k+2)(k+3)$:

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

Simplify the expression inside the parentheses:

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

Thus, the expression becomes:

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

This matches the right-hand side of the equation for $n = k+1$, completing the inductive step.

Conclusion:

By the principle of mathematical induction, the formula holds for all $n \geq 1$.

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:

1. What is the principle of mathematical induction and why is it used?
2. How can you identify patterns in a sequence that lead to an inductive proof?
3. What are common mistakes when applying the inductive step?
4. How can this approach be applied to other summation problems?
5. What is the significance of factoring expressions in proofs like this?

Tip: When proving by induction, always verify your base case carefully and ensure the inductive step follows logically.