Sure, let's dive into the given problem step by step.
We are asked to determine which of the given options matches the expression [tex]\((n + 1)^2 - n^2\)[/tex] for every natural number [tex]\( n \)[/tex].
### Step-by-Step Solution
#### Step 1: Expand the Expression [tex]\((n + 1)^2 - n^2\)[/tex]
First, we need to expand and simplify the expression [tex]\((n + 1)^2 - n^2\)[/tex].
[tex]\[
(n + 1)^2 = n^2 + 2n + 1
\][/tex]
So, the expression [tex]\((n + 1)^2 - n^2\)[/tex] becomes:
[tex]\[
(n^2 + 2n + 1) - n^2
\][/tex]
#### Step 2: Simplify the Expression
Next, we combine like terms:
[tex]\[
n^2 + 2n + 1 - n^2 = 2n + 1
\][/tex]
So, the simplified expression is:
[tex]\[
2n + 1
\][/tex]
#### Step 3: Compare with the Given Options
Let's now compare this simplified expression with the given answer choices to see which one matches:
1. [tex]\( n - (n + 1) \)[/tex]
[tex]\[
= n - n - 1
= -1
\][/tex]
2. [tex]\( (n + 1) - n \)[/tex]
[tex]\[
= n + 1 - n
= 1
\][/tex]
3. [tex]\( (n - 1) + n \)[/tex]
[tex]\[
= n - 1 + n
= 2n - 1
\][/tex]
4. [tex]\( (n + 1) + n \)[/tex]
[tex]\[
= n + 1 + n
= 2n + 1
\][/tex]
Comparing each of these options with our simplified expression [tex]\( 2n + 1 \)[/tex], it's clear that option (d) matches:
[tex]\[
(n + 1) + n = 2n + 1
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(n + 1) + n}
\][/tex]
This corresponds to option [tex]\( \boxed{(d)} \)[/tex].
