Let `Sn = n^2 + 20n + 12` where n is a positive integer. What is the sum of all possible values of n for which `S_(n)` is a perfect square ?

Answer

To solve the problem, we need to find the values of \( n \) such that \( S_n = n^2 + 20n + 12 \) is a perfect square. We will denote the perfect square as \( k^2 \) for some integer \( k \).

### Step 1: Set up the equation
We start with the equation:
\[ S_n = n^2 + 20n + 12 = k^2 \]
Rearranging gives:
\[ n^2 + 20n + (12 - k^2) = 0 \]

### Step 2: Use the discriminant
For \( n \) to be an integer, the discriminant of this quadratic equation must be a perfect square. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = 20^2 - 4 \cdot 1 \cdot (12 - k^2) = 400 - 48 + 4k^2 = 352 + 4k^2 \]
We want \( D \) to be a perfect square, say \( m^2 \):
\[ m^2 = 4k^2 + 352 \]

### Step 3: Rearranging the equation
We can rearrange this to:
\[ m^2 - 4k^2 = 352 \]
This can be factored as:
\[ (m - 2k)(m + 2k) = 352 \]

### Step 4: Factor 352
Next, we find the factor pairs of 352. The factors of 352 are:
1. \( 1 \times 352 \)
2. \( 2 \times 176 \)
3. \( 4 \times 88 \)
4. \( 8 \times 44 \)
5. \( 11 \times 32 \)
6. \( 16 \times 22 \)

### Step 5: Solve for \( n \)
For each factor pair \( (a, b) \), we set:
\[ m - 2k = a \quad \text{and} \quad m + 2k = b \]
Adding these gives:
\[ 2m = a + b \quad \Rightarrow \quad m = \frac{a + b}{2} \]
Subtracting gives:
\[ 4k = b - a \quad \Rightarrow \quad k = \frac{b - a}{4} \]

### Step 6: Check each factor pair
We need \( m \) and \( k \) to be integers. We will check each factor pair:

1. **For \( (1, 352) \)**:
- \( m = \frac{1 + 352}{2} = 176.5 \) (not an integer)

2. **For \( (2, 176) \)**:
- \( m = \frac{2 + 176}{2} = 89 \)
- \( k = \frac{176 - 2}{4} = 43.5 \) (not an integer)

3. **For \( (4, 88) \)**:
- \( m = \frac{4 + 88}{2} = 46 \)
- \( k = \frac{88 - 4}{4} = 21 \) (valid)

4. **For \( (8, 44) \)**:
- \( m = \frac{8 + 44}{2} = 26 \)
- \( k = \frac{44 - 8}{4} = 9 \) (valid)

5. **For \( (11, 32) \)**:
- \( m = \frac{11 + 32}{2} = 21.5 \) (not an integer)

6. **For \( (16, 22) \)**:
- \( m = \frac{16 + 22}{2} = 19 \)
- \( k = \frac{22 - 16}{4} = 1.5 \) (not an integer)

### Step 7: Calculate possible values of \( n \)
From valid pairs:
1. For \( k = 21 \):
\[ D = 352 + 4(21^2) = 352 + 1764 = 2116 \quad \Rightarrow \quad n = \frac{-20 \pm \sqrt{2116}}{2} \]
\[ \sqrt{2116} = 46 \quad \Rightarrow \quad n = \frac{-20 \pm 46}{2} \Rightarrow n = 13 \text{ or } n = -33 \text{ (discard negative)} \]

2. For \( k = 9 \):
\[ D = 352 + 4(9^2) = 352 + 324 = 676 \quad \Rightarrow \quad n = \frac{-20 \pm \sqrt{676}}{2} \]
\[ \sqrt{676} = 26 \quad \Rightarrow \quad n = \frac{-20 \pm 26}{2} \Rightarrow n = 3 \text{ or } n = -23 \text{ (discard negative)} \]

### Step 8: Sum of all possible values of \( n \)
The possible values of \( n \) are \( 13 \) and \( 3 \). Thus, the sum is:
\[ 13 + 3 = 16 \]

### Final Answer
The sum of all possible values of \( n \) for which \( S_n \) is a perfect square is \( \boxed{16} \).