To find the 10th term of the sequence \(1, -9, 81, \ldots\), we first need to determine the pattern in the sequence.

### Step 1: Identify the Pattern

The given sequence is:
\[
1, -9, 81, \ldots
\]

Looking at the pattern, each term seems to be multiplying the previous term by \(-9\):

\[
\begin{align*}
\text{First term:} & \quad 1 \\
\text{Second term:} & \quad 1 \times (-9) = -9 \\
\text{Third term:} & \quad -9 \times (-9) = 81 \\
\end{align*}
\]

The sequence is a **geometric progression** where the first term \(a = 1\) and the common ratio \(r = -9\).

### Step 2: Find the General Formula

The general formula for the \(n\)-th term of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]

### Step 3: Calculate the 10th Term

To find the 10th term (\(a_{10}\)):
\[
a_{10} = 1 \cdot (-9)^{10-1} = (-9)^9
\]

Now, compute \((-9)^9\):

\[
(-9)^9 = -(9^9)
\]

Calculate \(9^9\):

\[
9^9 = 387,420,489
\]

Thus,

\[
(-9)^9 = -387,420,489
\]

### Conclusion

The 10th term of the sequence is:
\[
\boxed{-387,420,489}
\]

Would you like more details or have any questions?

### Related Questions

1. What would be the 15th term of this sequence?
2. How do you determine if a sequence is geometric or arithmetic?
3. Can you find the sum of the first 10 terms of this geometric sequence?
4. What happens to the sign of the terms in a geometric sequence with a negative common ratio?
5. How would you calculate the \(n\)-th term if the first term is not 1 but some other number?

### Tip

Always check the common ratio in a geometric sequence; if it's negative, the signs of the terms alternate.

Learn how to find the 10th term of the geometric sequence 1, -9, 81, ..., with a detailed step-by-step solution. This sequence involves a negative common ratio and requires the use of the general formula for the nth term of a geometric sequence. Suitable for students in Grades 10-12.

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