Alright, let's solve this step by step.
We need to evaluate [tex]\((n+1)^2 - n^2\)[/tex] and compare it with the expressions given in the options.
### Step 1: Expand and Simplify [tex]\((n+1)^2 - n^2\)[/tex]
First, let's expand [tex]\((n+1)^2\)[/tex]:
[tex]\[ (n+1)^2 = n^2 + 2n + 1 \][/tex]
Next, let's subtract [tex]\(n^2\)[/tex] from this expanded form:
[tex]\[ (n^2 + 2n + 1) - n^2 \][/tex]
Simplify the expression:
[tex]\[ n^2 + 2n + 1 - n^2 = 2n + 1 \][/tex]
So, [tex]\((n+1)^2 - n^2\)[/tex] simplifies to [tex]\(2n + 1\)[/tex].
### Step 2: Evaluate Each Option
Now let's evaluate the expressions given in the options to see which one matches [tex]\(2n + 1\)[/tex].
#### Option (a) [tex]\( n - (n+1)\)[/tex]
[tex]\[ n - (n+1) = n - n - 1 = -1 \][/tex]
This simplifies to [tex]\(-1\)[/tex].
#### Option (b) [tex]\((n+1) - n\)[/tex]
[tex]\[ (n+1) - n = n + 1 - n = 1 \][/tex]
This simplifies to [tex]\(1\)[/tex].
#### Option (d) [tex]\((n+1) + n\)[/tex]
[tex]\[ (n+1) + n = n + 1 + n = 2n + 1 \][/tex]
This simplifies to [tex]\(2n + 1\)[/tex].
### Step 3: Compare With [tex]\((n+1)^2 - n^2\)[/tex]
We found that [tex]\((n+1)^2 - n^2\)[/tex] is [tex]\(2n + 1\)[/tex].
- Option (a) is [tex]\(-1\)[/tex], which does not match.
- Option (b) is [tex]\(1\)[/tex], which does not match.
- Option (d) is [tex]\(2n + 1\)[/tex], which matches.
### Final Answer
The expression [tex]\((n+1)^2 - n^2\)[/tex] equals [tex]\(2n + 1\)[/tex], which matches with option (d).
Therefore, for every natural number [tex]\(n\)[/tex],
[tex]\[ (n+1)^2 - n^2 = 2n + 1 \][/tex]
and the correct choice is:
[tex]\[ \textbf{(d)} \ (n+1) + n \][/tex]
