To find the next two terms of the sequence $-5, 10, -20$, let's analyze the pattern.

Step 1: Determine the ratio between consecutive terms.

• The ratio between the second term and the first term is: $\frac{10}{-5} = -2$
• The ratio between the third term and the second term is: $\frac{-20}{10} = -2$

Step 2: Confirm the sequence follows a geometric progression.

Since the ratio is constant ($-2$), this is a geometric sequence with a common ratio $r = -2$.

Step 3: Calculate the next two terms.

The formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Using $a_1 = -5$ and $r = -2$:

1. Fourth term ($a_4$): $a_4 = a_3 \cdot r = -20 \cdot (-2) = 40$
2. Fifth term ($a_5$): $a_5 = a_4 \cdot r = 40 \cdot (-2) = -80$

Final Answer:

The next two terms are: $40, -80$

Would you like a detailed explanation of geometric sequences, or do you have any questions?

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Related Questions:

1. What is the general formula for this sequence?
2. How does the common ratio affect the sequence?
3. What are the first ten terms of this sequence?
4. Can the sum of the first $n$ terms of this sequence be calculated? If so, how?
5. How does a geometric sequence differ from an arithmetic sequence?

Tip:

In a geometric sequence, always check the ratio between consecutive terms to verify consistency before proceeding.