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.