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.