The given sequence is:

50, -10, 2

To find the nth term of this sequence, we'll first determine if it follows a recognizable pattern, such as an arithmetic or geometric sequence.

### Step 1: Find the common differences

We subtract each term from the next:

- \( -10 - 50 = -60 \)
- \( 2 - (-10) = 12 \)

Since the differences are not constant, the sequence is not arithmetic.

### Step 2: Assume a quadratic form

We assume the nth term follows the form:
\[
T_n = an^2 + bn + c
\]
We can now solve for \(a\), \(b\), and \(c\) using the first three terms.

For \(n = 1\) (first term):
\[
T_1 = a(1)^2 + b(1) + c = 50 \implies a + b + c = 50 \quad (1)
\]

For \(n = 2\) (second term):
\[
T_2 = a(2)^2 + b(2) + c = -10 \implies 4a + 2b + c = -10 \quad (2)
\]

For \(n = 3\) (third term):
\[
T_3 = a(3)^2 + b(3) + c = 2 \implies 9a + 3b + c = 2 \quad (3)
\]

### Step 3: Solve the system of equations

We now have the system of equations:
1. \( a + b + c = 50 \)
2. \( 4a + 2b + c = -10 \)
3. \( 9a + 3b + c = 2 \)

Let's solve this system.

1. Subtract equation (1) from equation (2):
\[
(4a + 2b + c) - (a + b + c) = -10 - 50
\]
\[
3a + b = -60 \quad (4)
\]

2. Subtract equation (2) from equation (3):
\[
(9a + 3b + c) - (4a + 2b + c) = 2 - (-10)
\]
\[
5a + b = 12 \quad (5)
\]

3. Subtract equation (4) from equation (5):
\[
(5a + b) - (3a + b) = 12 - (-60)
\]
\[
2a = 72 \quad \implies a = 36
\]

4. Substitute \(a = 36\) into equation (4):
\[
3(36) + b = -60
\]
\[
108 + b = -60 \quad \implies b = -168
\]

5. Substitute \(a = 36\) and \(b = -168\) into equation (1):
\[
36 - 168 + c = 50
\]
\[
c = 50 + 168 - 36 = 182
\]

### Step 4: Write the nth term

Now that we have \(a = 36\), \(b = -168\), and \(c = 182\), the nth term is:
\[
T_n = 36n^2 - 168n + 182
\]

This is the formula for the nth term of the given sequence.

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Do you want further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you verify if a sequence follows a quadratic pattern?
2. What are the differences between arithmetic and geometric sequences?
3. How do you solve a system of linear equations in three variables?
4. Can you derive the nth term for a geometric sequence instead?
5. What happens if a sequence follows a cubic pattern?

**Tip:** When dealing with non-linear sequences, try assuming a polynomial expression for the nth term, then solve using the known values.

Learn how to find the nth term for the sequence 50, −10, 2 by solving a system of equations. This step-by-step solution covers quadratic sequences, solving equations, and algebraic concepts suitable for Grades 9-11 students.

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