To determine if the sequence $-1, -5, -25, -125$ is geometric, we need to check if there is a constant ratio between consecutive terms. If so, we'll find the common ratio, the 8th term, and the explicit formula.

Step 1: Check for a constant ratio

The ratio between consecutive terms is calculated as: $r = \frac{a_{n+1}}{a_n}$

• Between the 1st and 2nd terms: $r = \frac{-5}{-1} = 5$

• Between the 2nd and 3rd terms: $r = \frac{-25}{-5} = 5$

• Between the 3rd and 4th terms: $r = \frac{-125}{-25} = 5$

Since the ratio $r = 5$ is constant, the sequence is geometric.

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Step 2: Find the explicit formula

For a geometric sequence, the $n$-th term is given by: $a_n = a_1 \cdot r^{n-1}$

Here, $a_1 = -1$ and $r = 5$. Substituting these values, the explicit formula becomes: $a_n = -1 \cdot 5^{n-1} = -5^{n-1}$

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Step 3: Calculate the 8th term

Using the explicit formula, we find the 8th term ($a_8$): $a_8 = -5^{8-1} = -5^7$

Calculate $5^7$: $5^7 = 78125 \quad \Rightarrow \quad a_8 = -78125$

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Final Results:

1. The sequence is geometric.
2. Common ratio: $r = 5$
3. Explicit formula: $a_n = -5^{n-1}$
4. 8th term: $a_8 = -78125$

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Related Questions:

1. What is the sum of the first $n$ terms of this geometric sequence?
2. How does the formula for geometric sequences differ for increasing versus decreasing sequences?
3. Can the common ratio $r$ be a fraction, and how does that affect the sequence?
4. How can we determine if a sequence is geometric from a given formula?
5. What is the explicit formula if the sequence starts from a different first term?

Tip:

For any geometric sequence, remember that the ratio $r$ can be positive or negative. A negative ratio alternates the sign of the terms!