insert -2 geometric means between 10 and -160

To insert two geometric means between 10 and -160, we need to determine three terms of a geometric sequence: the first term (\(a_1\)), the two geometric means (\(a_2\) and \(a_3\)), and the fourth term (\(a_4\)).

### Step-by-Step Solution:

1. **Identify the Known Terms:**
   - \(a_1 = 10\) (first term)
   - \(a_4 = -160\) (fourth term)

2. **Set Up the General Formula for the Geometric Sequence:**
   In a geometric sequence, each term is found by multiplying the previous term by a common ratio \(r\). The general term of a geometric sequence is given by:
   \[
   a_n = a_1 \cdot r^{n-1}
   \]

   Therefore:
   \[
   a_2 = a_1 \cdot r
   \]
   \[
   a_3 = a_1 \cdot r^2
   \]
   \[
   a_4 = a_1 \cdot r^3
   \]

3. **Apply the Known Values:**
   Since \(a_4 = -160\) and \(a_1 = 10\), we have:
   \[
   -160 = 10 \cdot r^3
   \]

4. **Solve for the Common Ratio \(r\):**
   \[
   r^3 = \frac{-160}{10} = -16
   \]
   \[
   r = \sqrt[3]{-16} = -2
   \]

5. **Determine the Geometric Means \(a_2\) and \(a_3\):**
   \[
   a_2 = a_1 \cdot r = 10 \cdot (-2) = -20
   \]
   \[
   a_3 = a_1 \cdot r^2 = 10 \cdot (-2)^2 = 10 \cdot 4 = 40
   \]

### Final Answer:
The two geometric means between 10 and -160 are **-20** and **40**.

Would you like more details or have any other questions? Here are some related questions you might find interesting:

1. What happens if you need to insert more than two geometric means?
2. How would you find the geometric means if the sequence included more terms?
3. Can a geometric sequence have both positive and negative terms? What are the implications?
4. What if the sequence started with a negative number? How would that affect the geometric means?
5. How do geometric sequences differ from arithmetic sequences in terms of their properties?

**Tip:** When working with geometric sequences, always check the sign of the common ratio carefully, as it significantly affects the sequence's behavior.

Learn how to insert two geometric means between 10 and -160 in a geometric sequence with a detailed step-by-step solution. This problem covers key concepts in geometric sequences and calculations involving the common ratio. Suitable for advanced high school students.

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