Soit pour n $$ \in\text{ N} $$ la propriété Pn : 1 + 2 + ... + n = n(n+1)/2 

The given problem involves proving the property $ P_n $ for $ n \in \mathbb{N} $:

\[
P_n : 1 + 2 + \cdots + n = \frac{n(n+1)}{2}
\]

This is a classical result, and we can prove it by **mathematical induction**.

### Step 1: Base Case $ n = 1 $

For $ n = 1 $, the left-hand side of the equation is simply:
\[
1 = 1
\]
The right-hand side is:
\[
\frac{1(1+1)}{2} = \frac{1 \times 2}{2} = 1
\]
Since both sides are equal, the base case holds true.

### Step 2: Inductive Hypothesis

Assume that the property holds for some $ n = k $, i.e.,
\[
1 + 2 + \cdots + k = \frac{k(k+1)}{2}
\]
This is our **inductive hypothesis**.

### Step 3: Inductive Step

We now need to prove that if the property holds for $ n = k $, then it also holds for $ n = k+1 $, i.e.,

\[
1 + 2 + \cdots + k + (k+1) = \frac{(k+1)((k+1) + 1)}{2}
\]

Using the inductive hypothesis, the left-hand side becomes:
\[
\left(1 + 2 + \cdots + k\right) + (k+1) = \frac{k(k+1)}{2} + (k+1)
\]
Now, factor out $ (k+1) $:
\[
\frac{k(k+1)}{2} + \frac{2(k+1)}{2} = \frac{k(k+1) + 2(k+1)}{2}
\]
\[
= \frac{(k+1)(k+2)}{2}
\]
This matches the right-hand side of the equation for $ n = k+1 $. Hence, the property holds for $ n = k+1 $.

### Conclusion

By the principle of mathematical induction, the property $ P_n : 1 + 2 + \cdots + n = \frac{n(n+1)}{2} $ is true for all $ n \in \mathbb{N} $.

Would you like more details or have any questions about this?

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Here are five related questions to deepen your understanding:
1. How does mathematical induction work, and why is it a valid proof technique?
2. What are some common patterns in sums, like the sum of squares or cubes?
3. How can we derive a closed-form formula for the sum of an arithmetic series?
4. What is the connection between this sum and the area of a triangle in geometry?
5. Can you prove this formula using other methods, such as a combinatorial argument?

**Tip**: Always verify the base case carefully when using induction—it ensures the foundation of the proof!

Soit pour n ∈ N la propriété Pn : 1 + 2 + ... + n = n(n+1)/2

This page provides a step-by-step proof of the summation formula 1 + 2 + ... + n = n(n+1)/2 using mathematical induction. Ideal for high school students in Grades 10-12, the solution covers arithmetic series and the principle of mathematical induction.

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