What is `((n + 2)! + (n + 1) (n -1)!)/((n +1) (n -1)!)` equal to ?

Options

A. 1
B. Always an odd integer
C. A perfect square
D. None of the above

Explanation

To solve the expression \(\frac{(n + 2)! + (n + 1)(n - 1)!}{(n + 1)(n - 1)!}\), we will simplify it step by step.

### Step 1: Rewrite the Factorials
First, we can rewrite \((n + 2)!\) in terms of \((n + 1)!\):
\[ (n + 2)! = (n + 2)(n + 1)(n)! \]
So, we can substitute this into the expression:
\[ \frac{(n + 2)(n + 1)(n)! + (n + 1)(n - 1)!}{(n + 1)(n - 1)!} \]

### Step 2: Factor Out Common Terms
Notice that \((n + 1)\) is a common factor in the numerator:
\[ = \frac{(n + 1) \left((n + 2)(n)! + (n - 1)!\right)}{(n + 1)(n - 1)!} \]
Now, we can cancel \((n + 1)\) from the numerator and the denominator (assuming \(n + 1 \neq 0\)):
\[ = \frac{(n + 2)(n)! + (n - 1)!}{(n - 1)!} \]

### Step 3: Simplify the Expression Further
Now, we can separate the terms in the numerator:
\[ = \frac{(n + 2)(n)!}{(n - 1)!} + \frac{(n - 1)!}{(n - 1)!} \]
The second term simplifies to 1:
\[ = \frac{(n + 2)(n)(n - 1)!}{(n - 1)!} + 1 \]
Now, we can cancel \((n - 1)!\):
\[ = (n + 2)n + 1 \]

### Step 4: Final Simplification
Now, we can expand the expression:
\[ = n^2 + 2n + 1 \]
This can be recognized as a perfect square:
\[ = (n + 1)^2 \]

### Conclusion
Thus, the expression simplifies to:
\[ \frac{(n + 2)! + (n + 1)(n - 1)!}{(n + 1)(n - 1)!} = (n + 1)^2 \]

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