ыщдму ершы зкщидуь гыштп mathematical induction

It seems like you're asking about **mathematical induction**. Let me explain what it is.

**Mathematical induction** is a method of mathematical proof typically used to prove that a statement holds for all natural numbers (or some infinite sequence). The basic idea is to prove the base case first, and then prove that if the statement holds for one number, it holds for the next one as well. 

### Steps of Mathematical Induction:
1. **Base Case:** Prove that the statement is true for the initial value, often \( n = 1 \) or \( n = 0 \).
   
2. **Inductive Step:** 
   - Assume the statement is true for \( n = k \) (this is called the *inductive hypothesis*).
   - Then, prove that if the statement holds for \( n = k \), it must also hold for \( n = k+1 \).

If both the base case and the inductive step are true, the statement is true for all \( n \) starting from the base case.

#### Example: Proving the sum of the first \( n \) natural numbers.
Let's prove that \( 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \) using induction.

1. **Base Case:**
   For \( n = 1 \):
   \[
   1 = \frac{1(1+1)}{2} = 1
   \]
   The statement is true for \( n = 1 \).

2. **Inductive Hypothesis:**
   Assume the statement is true for \( n = k \), i.e.,
   \[
   1 + 2 + 3 + \dots + k = \frac{k(k+1)}{2}.
   \]

3. **Inductive Step:**
   We need to show that the statement holds for \( n = k+1 \):
   \[
   1 + 2 + 3 + \dots + k + (k+1) = \frac{(k+1)((k+1)+1)}{2}.
   \]
   Start with the inductive hypothesis:
   \[
   1 +

Learn the steps of mathematical induction and how to use it to prove formulas involving natural numbers. This example shows how to prove the sum of the first n natural numbers, following the base case and inductive step. Suitable for high school students in Grades 10-12.

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