Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


–8 , –32 , –128 , ...


Write your answer using decimals and integers.

an = _(_)^ n^ -^ 1

Answer 1

To find the equation that describes the given sequence \(-8, -32, -128, \ldots\), we first need to identify the pattern in the terms.

1. Let's observe the sequence:
- First term (\(a_1\)): \(-8\)
- Second term (\(a_2\)): \(-32\)
- Third term (\(a_3\)): \(-128\)

2. Now let's try to find a common ratio by dividing consecutive terms:
- \(\frac{a_2}{a_1} = \frac{-32}{-8} = 4\)
- \(\frac{a_3}{a_2} = \frac{-128}{-32} = 4\)

This shows that the sequence is geometric with a common ratio of \(4\).

3. The first term can also be expressed as \(-8\), and since this is a geometric sequence, we can represent any term in the sequence using the formula for a geometric series:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term.

4. Plugging in the values:
- \(a_1 = -8\)
- \(r = 4\)

Thus, the formula becomes:
\[
a_n = -8 \cdot 4^{(n-1)}
\]

Therefore, the final equation that describes the sequence is:
\[
a_n = -8(4)^{n-1}
\]