To find the 10th term of a geometric sequence, we use the formula:

$a_n = a_1 \cdot r^{n-1}$

Where:

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

Step 1: Determine $a_1$ and $r$

The first term ($a_1$) is $-4.5$.
The common ratio ($r$) is calculated as:

$r = \frac{a_2}{a_1} = \frac{-13.5}{-4.5} = 3$

Step 2: Substitute into the formula

We want the 10th term ($a_{10}$):

$a_{10} = a_1 \cdot r^{10-1} = -4.5 \cdot 3^{9}$

Step 3: Calculate $3^9$

$3^9 = 19683$

Substitute this back:

$a_{10} = -4.5 \cdot 19683 = -88573.5$

Final Answer:

The 10th term is:

$\boxed{-88573.5}$

Would you like further clarification or a breakdown of the calculations? 😊

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Related Questions:

1. How is the common ratio calculated in geometric sequences?
2. What are the key differences between geometric and arithmetic sequences?
3. How can we derive the sum of the first $n$ terms in a geometric sequence?
4. Can geometric sequences include positive and negative terms?
5. How do we solve for $n$ if a specific term value is given?

Tip:

When working with large exponents, breaking the computation into smaller steps can reduce errors!