Questions: (x+1)!+(x-1)!/x!

Solution Steps

We start with the expression \(\frac{(x+1)! + (x-1)!}{x!}\). To simplify this, we can express \((x+1)!\) and \((x-1)!\) in terms of \(x!\): \[ (x+1)! = (x+1) \cdot x! \] \[ (x-1)! = \frac{x!}{x} \quad \text{(for } x \geq 1\text{)} \] Substituting these into the expression gives: \[ \frac{(x+1) \cdot x! + \frac{x!}{x}}{x!} \]

Next, we can factor \(x!\) out of the numerator: \[ \frac{x! \left((x+1) + \frac{1}{x}\right)}{x!} \] This simplifies to: \[ (x+1) + \frac{1}{x} \]

Now, we substitute \(x = 5\) into the simplified expression: \[ 5 + 1 + \frac{1}{5} = 6 + 0.2 = 6.2 \]

Final Answer