Let's denote the first term of the arithmetic progression (AP) as $a$, and the common difference as $d$.

Step 1: Use the information about the 13th term

The general formula for the $n$-th term of an arithmetic progression is:

$T_n = a + (n-1) d$

For the 13th term, we know that:

$T_{13} = a + (13-1) d = a + 12d$

We are told that $T_{13} = 12$, so:

$a + 12d = 12 \quad \text{(Equation 1)}$

Step 2: Use the information about the sum of the first 30 terms

The sum of the first $n$ terms of an arithmetic progression is given by the formula:

$S_n = \frac{n}{2} \left( 2a + (n-1) d \right)$

For $n = 30$, we are told that $S_{30} = -15$, so:

$S_{30} = \frac{30}{2} \left( 2a + (30-1) d \right) = 15 \left( 2a + 29d \right) = -15$

Simplifying:

$2a + 29d = -1 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

We now have the system of two equations:

$a + 12d = 12 \quad \text{(Equation 1)}$ $2a + 29d = -1 \quad \text{(Equation 2)}$

Solving Equation 1 for $a$:

From Equation 1:

$a = 12 - 12d$

Substitute into Equation 2:

Substitute $a = 12 - 12d$ into Equation 2:

$2(12 - 12d) + 29d = -1$ $24 - 24d + 29d = -1$ $24 + 5d = -1$ $5d = -25$ $d = -5$

Find $a$:

Now that we know $d = -5$, substitute this into $a = 12 - 12d$:

$a = 12 - 12(-5) = 12 + 60 = 72$

Step 4: Find the sum of the first 50 terms

Now we need to find the sum of the first 50 terms, $S_{50}$. Using the sum formula:

$S_{50} = \frac{50}{2} \left( 2a + (50-1) d \right)$ $S_{50} = 25 \left( 2(72) + 49(-5) \right)$ $S_{50} = 25 \left( 144 - 245 \right)$ $S_{50} = 25 \times (-101)$ $S_{50} = -2525$

Final Answer:

The sum of the first 50 terms is $\boxed{-2525}$.

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Would you like further clarification or have any questions?

Here are 5 related questions to explore further:

1. How would the sum change if the common difference was positive?
2. How do you find the sum of an infinite arithmetic progression?
3. What happens to the sum of an arithmetic progression if the common difference is 0?
4. How do you find the sum of an arithmetic progression when the last term is known, instead of the total number of terms?
5. What is the general method to derive the formula for the $n$-th term in an arithmetic progression?

Tip: Always check if the arithmetic progression's sum involves a negative or positive value, as this can significantly impact the behavior of the sequence, especially in practical applications!