Madyson borrows $$ \$ 1,200 $$ from her father to purchase airplane tickets. The first week after borrowing the money, Madyson repays her father $$ \$ 50 $$. Each week after that, she is able to repay her father $$ \$ 2 $$ more than the previous week, except in the last week, when she finishes repaying the loan. How much is Madyson's final payment? $$ \$ 78 $$ $$ \$ 82 $$ $$ \$ 84 $$ $$ \$ 88 $$

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To solve the problem, we need to determine how much Madyson's final payment is after she has been repaying her father in increasing amounts each week.

1. **Identify the known conditions:**
   - Initial loan amount: $$ P = 1200 $$
   - First week's payment: $$ p_1 = 50 $$
   - Each subsequent week's payment increases by $$ 2 $$: $$ p_n = p_1 + 2(n - 1) $$

2. **Set up the equation for total payments:**
   - The total amount paid after $$ n $$ weeks can be expressed as:
   $$
   \text{Total Payments} = p_1 + p_2 + p_3 + \ldots + p_n
   $$
   - The payment for week $$ n $$ is:
   $$
   p_n = 50 + 2(n - 1) = 50 + 2n - 2 = 2n + 48
   $$
   - The total payment can be expressed as:
   $$
   \text{Total Payments} = 50 + (50 + 2) + (50 + 4) + \ldots + (2n + 48)
   $$

3. **Calculate the total payments:**
   - The total payments can be simplified using the formula for the sum of an arithmetic series:
   $$
   \text{Total Payments} = n \cdot \text{average payment}
   $$
   - The average payment can be calculated as:
   $$
   \text{Average Payment} = \frac{p_1 + p_n}{2} = \frac{50 + (2n + 48)}{2} = \frac{2n + 98}{2} = n + 49
   $$
   - Therefore, the total payments become:
   $$
   \text{Total Payments} = n(n + 49)
   $$

4. **Set the total payments equal to the loan amount:**
   $$
   n(n + 49) = 1200
   $$
   - Rearranging gives us a quadratic equation:
   $$
   n^2 + 49n - 1200 = 0
   $$

5. **Solve the quadratic equation:**
   We can use the quadratic formula:
   $$
   n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   where $$ a = 1, b = 49, c = -1200 $$.

First, calculate the discriminant:
   $$
   b^2 - 4ac = 49^2 - 4 \cdot 1 \cdot (-1200) = 2401 + 4800 = 7201
   $$

Now, calculate $$ n $$:
   $$
   n = \frac{-49 \pm \sqrt{7201}}{2}
   $$

Let's calculate $$ \sqrt{7201} $$ and then find $$ n $$.

6. **Calculate $$ n $$:**
   $$
   \sqrt{7201} \approx 84.85
   $$
   Thus,
   $$
   n = \frac{-49 + 84.85}{2} \approx \frac{35.85}{2} \approx 17.925
   $$
   Since $$ n $$ must be a whole number, we round down to $$ n = 17 $$.

7. **Calculate the final payment:**
   Now, we can find the final payment:
   $$
   p_n = 2n + 48 = 2(17) + 48 = 34 + 48 = 82
   $$

Thus, Madyson's final payment is $$ \boxed{82} $$.
Madyson's final payment is \$82.

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