To find the equation that describes the given sequence: 8, 40, 200, ..., we first need to analyze the pattern in the sequence.
Let's look at the sequence step-by-step:
- First Term (\(n=1\)): \(a_1 = 8\)
- Second Term (\(n=2\)): \(a_2 = 40\)
- Third Term (\(n=3\)): \(a_3 = 200\)
Next, we can identify how the terms relate to each other.
Calculating the ratio of consecutive terms:
- \( \frac{a_2}{a_1} = \frac{40}{8} = 5 \)
- \( \frac{a_3}{a_2} = \frac{200}{40} = 5 \)
This suggests that each term is 5 times the previous term.
Thus, we can determine that this can be represented as:
\[ a_n = a_{n-1} \cdot 5 \]
Starting with \(a_1 = 8\), we can express this in a formula form:
This suggests a geometric sequence. A geometric sequence can be expressed as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Where:
- \(a_1\) is the first term,
- \(r\) is the common ratio, and
- \(n\) is the term number.
Given \(a_1 = 8\) and \(r = 5\), we can now write our equation for the \(n\)-th term:
\[ a_n = 8 \cdot 5^{(n-1)} \]
So, the equation that describes the sequence is:
\[ a_n = 8 \cdot 5^{(n-1)} \]