To find a formula for the $n$th term of the sequence $18, -12, 8$, we can start by identifying the pattern in the sequence.

1. Identify the terms:

   • $a_1 = 18$
   • $a_2 = -12$
   • $a_3 = 8$

2. Calculate the differences between consecutive terms:

   • $a_2 - a_1 = -12 - 18 = -30$
   • $a_3 - a_2 = 8 - (-12) = 20$

3. Check for a second difference:

   • The first differences: $-30, 20$
   • Second difference: $20 - (-30) = 50$

Since the differences do not form a consistent pattern, it appears this sequence might not be linear. However, we can find a polynomial that fits these points.

Assuming a quadratic form: $a_n = An^2 + Bn + C$

Using the first three terms, we can set up a system of equations:

• For $n = 1$: $A(1)^2 + B(1) + C = 18$ → $A + B + C = 18$ (1)
• For $n = 2$: $A(2)^2 + B(2) + C = -12$ → $4A + 2B + C = -12$ (2)
• For $n = 3$: $A(3)^2 + B(3) + C = 8$ → $9A + 3B + C = 8$ (3)

Now we solve this system of equations.

Subtract (1) from (2): $(4A + 2B + C) - (A + B + C) = -12 - 18$ $3A + B = -30 \quad \text{(4)}$

Subtract (2) from (3): $(9A + 3B + C) - (4A + 2B + C) = 8 - (-12)$ $5A + B = 20 \quad \text{(5)}$

Now subtract (4) from (5): $(5A + B) - (3A + B) = 20 - (-30)$ $2A = 50 \quad \Rightarrow \quad A = 25$

Substituting $A = 25$ back into (4): $3(25) + B = -30 \quad \Rightarrow \quad 75 + B = -30 \quad \Rightarrow \quad B = -105$

Now substitute $A$ and $B$ back into (1): $25 - 105 + C = 18 \quad \Rightarrow \quad C = 18 + 105 - 25 = 98$

Thus, the formula for the $n$th term is: $a_n = 25n^2 - 105n + 98$

Summary

The formula for the $n$th term of the sequence is: $a_n = 25n^2 - 105n + 98$

Would you like any more details or have specific questions? Here are some related questions you might consider:

1. How can you find the sum of the first $n$ terms of this sequence?
2. What is the graphical representation of this quadratic function?
3. How does changing the coefficients $A$, $B$, and $C$ affect the shape of the parabola?
4. Can you find the roots of the quadratic equation derived from this sequence?
5. What would the sequence look like if we altered the first term to a different value?

Tip: Always check the behavior of a polynomial by analyzing its roots and how it behaves as $n$ increases.