Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

8 , 40 , 200 , ...

Write your answer using decimals and integers.

an =
(
)n - 1

1 answer

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:

  1. First Term (\(n=1\)): \(a_1 = 8\)
  2. Second Term (\(n=2\)): \(a_2 = 40\)
  3. 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:

\[ a_n = 8 \cdot 5^{(n-1)} \]