If `.^(n-1)P_3 :^(n+1)P_3 = 5 : 12,` find `n`.

Answer

To solve the problem, we need to find the value of \( n \) given the ratio \( \frac{(n-1)P_3}{(n+1)P_3} = \frac{5}{12} \).

### Step-by-Step Solution:

1. **Understand the formula for permutations**:
The formula for permutations is given by:
\[ nPr = \frac{n!}{(n-r)!} \]
For our case, we have \( r = 3 \).

2. **Write the expressions for \( (n-1)P_3 \) and \( (n+1)P_3 \)**:
\[ (n-1)P_3 = \frac{(n-1)!}{(n-1-3)!} = \frac{(n-1)!}{(n-4)!} \]
\[ (n+1)P_3 = \frac{(n+1)!}{(n+1-3)!} = \frac{(n+1)!}{(n-2)!} \]

3. **Set up the ratio**:
\[ \frac{(n-1)P_3}{(n+1)P_3} = \frac{\frac{(n-1)!}{(n-4)!}}{\frac{(n+1)!}{(n-2)!}} = \frac{(n-1)! \cdot (n-2)!}{(n+1)! \cdot (n-4)!} \]

4. **Simplify the ratio**:
\[ = \frac{(n-1)! \cdot (n-2)!}{(n+1) \cdot n \cdot (n-1)! \cdot (n-4)!} \]
Canceling \( (n-1)! \) from numerator and denominator:
\[ = \frac{(n-2)!}{(n+1) \cdot n \cdot (n-4)!} \]

5. **Further simplify**:
\[ = \frac{(n-2)(n-3)(n-4)!}{(n+1) \cdot n \cdot (n-4)!} \]
Cancel \( (n-4)! \):
\[ = \frac{(n-2)(n-3)}{(n+1) \cdot n} \]

6. **Set the equation equal to the given ratio**:
\[ \frac{(n-2)(n-3)}{(n+1) \cdot n} = \frac{5}{12} \]

7. **Cross-multiply to eliminate the fraction**:
\[ 12(n-2)(n-3) = 5(n+1)n \]

8. **Expand both sides**:
- Left side:
\[ 12(n^2 - 5n + 6) = 12n^2 - 60n + 72 \]
- Right side:
\[ 5(n^2 + n) = 5n^2 + 5n \]

9. **Set the equation to zero**:
\[ 12n^2 - 60n + 72 = 5n^2 + 5n \]
Rearranging gives:
\[ 12n^2 - 5n^2 - 60n - 5n + 72 = 0 \]
\[ 7n^2 - 65n + 72 = 0 \]

10. **Use the quadratic formula to find \( n \)**:
The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 7, b = -65, c = 72 \):
\[ n = \frac{65 \pm \sqrt{(-65)^2 - 4 \cdot 7 \cdot 72}}{2 \cdot 7} \]
\[ = \frac{65 \pm \sqrt{4225 - 2016}}{14} \]
\[ = \frac{65 \pm \sqrt{2209}}{14} \]
\[ = \frac{65 \pm 47}{14} \]

11. **Calculate the two possible values for \( n \)**:
- First value:
\[ n = \frac{112}{14} = 8 \]
- Second value:
\[ n = \frac{18}{14} \approx 1.28 \]

12. **Determine the valid solution**:
Since \( n \) must be a natural number, the only valid solution is:
\[ n = 8 \]

### Final Answer:
The value of \( n \) is \( 8 \).