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The task in your image asks to show that the following numbers are odd for a natural number \( n \):

1. \( (2n + 2)^2 - (2n + 1)^2 \)
2. \( n^3 - n + 1 \)
3. \( n^2 + 13n + 17 \)

Let's solve these one by one:

### 1. \( (2n + 2)^2 - (2n + 1)^2 \)

Using the identity for the difference of squares:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, \( a = 2n + 2 \) and \( b = 2n + 1 \). So,

\[
(2n + 2)^2 - (2n + 1)^2 = [(2n + 2) - (2n + 1)] \cdot [(2n + 2) + (2n + 1)]
\]

Simplifying:

\[
= (1) \cdot (4n + 3) = 4n + 3
\]

Since \( 4n + 3 \) is clearly of the form \( 2k + 1 \), it is an odd number.

### 2. \( n^3 - n + 1 \)

We need to determine if \( n^3 - n + 1 \) is odd for all \( n \).

- For \( n = 1 \):
  \[
  n^3 - n + 1 = 1^3 - 1 + 1 = 1
  \]
  which is odd.

- For \( n = 2 \):
  \[
  n^3 - n + 1 = 2^3 - 2 + 1 = 8 - 2 + 1 = 7
  \]
  which is odd.

It can be observed that for any natural number \( n \), the expression \( n^3 - n \) is always even because both \( n^3 \) and \( n \) have the same parity (both are either odd or even), so their difference is always even. Adding 1 makes the expression \( n^3 - n + 1 \) always odd.

### 3. \( n^2 + 13n + 17 \)

We need to check if \( n^2 + 13n + 17 \) is always odd.

- For \( n = 1 \):
  \[
  n^2 + 13n + 17 = 1^2 + 13(1) + 17 = 1 + 13 + 17 = 31
  \]
  which is odd.

- For \( n = 2 \):
  \[
  n^2 + 13n + 17 = 2^2 + 13(2) + 17 = 4 + 26 + 17 = 47
  \]
  which is odd.

For any natural number \( n \), the parity of \( n^2 + 13n \) follows that both \( n^2 \) and \( 13n \) have the same parity (both are either even or odd), so their sum is always even. Adding 17, an odd number, makes the entire expression odd.

---

In conclusion, all three expressions are indeed odd for any natural number \( n \).

Would you like further details on any step?

### Related Questions:
1. How can we extend the difference of squares formula to solve similar problems?
2. What patterns emerge when checking the parity of polynomial expressions?
3. Can the expression \( n^3 - n + 1 \) ever be even for some value of \( n \)?
4. How can you systematically determine whether a complex algebraic expression is odd or even?
5. What role does the difference of squares play in number theory?

**Tip:** The difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \) is a useful tool in simplifying and solving many algebraic expressions.

Soit n entier naturel. Montre que les nombres suivants sont impairs: 1. (2n + 2)^2 - (2n + 1)^2 2. n^3 - n + 1 3. n^2 + 13n + 17

This problem involves proving that expressions like (2n + 2)^2 - (2n + 1)^2 and n^3 - n + 1 are odd for any natural number n. Learn about the difference of squares and parity to solve this problem, suitable for students in Grades 8-10.

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