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.