The problem provided asks to find the first term and general expression of a geometric sequence. Here’s a breakdown of the given information:

• The third term $a_3 = -18$
• The sixth term $a_6 = 9,216$

In a geometric sequence, each term is given by: $a_n = a_1 \cdot r^{n-1}$ where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $r$ is the common ratio.

Step 1: Set up equations for $a_3$ and $a_6$

From the formula for the $n$-th term: $a_3 = a_1 \cdot r^2 = -18$ $a_6 = a_1 \cdot r^5 = 9216$

Step 2: Divide the two equations

Dividing the second equation by the first to eliminate $a_1$: $\frac{a_6}{a_3} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = r^3$ This simplifies to: $\frac{9216}{-18} = r^3$ $r^3 = -512$ Taking the cube root of both sides: $r = -8$

Step 3: Solve for $a_1$

Now substitute $r = -8$ into one of the original equations, say $a_3 = a_1 \cdot r^2 = -18$: $a_1 \cdot (-8)^2 = -18$ $a_1 \cdot 64 = -18$ $a_1 = \frac{-18}{64} = \frac{-9}{32}$

Step 4: General expression for the $n$-th term

The general formula for the $n$-th term is: $a_n = a_1 \cdot r^{n-1}$ Substituting $a_1 = \frac{-9}{32}$ and $r = -8$, we get: $a_n = \frac{-9}{32} \cdot (-8)^{n-1}$

Would you like further clarification or details on any step?

Here are 5 related questions to explore:

1. How can you verify if the calculated terms match the original sequence?
2. What would happen if the third term and sixth term were different?
3. How do you derive the formula for the sum of a geometric series?
4. How does the value of $r$ affect the behavior of the sequence?
5. What are some real-life applications of geometric sequences?

Tip: Always check for consistency by substituting your values back into the original sequence to ensure they satisfy all conditions.