To determine the explicit formula for the given sequence [tex]\(5, 10, 20, 40, 80, 160, \ldots\)[/tex], let's follow a clear, step-by-step approach.
1. Identify the Pattern:
Let's examine the ratio between consecutive terms to identify if this might be a geometric sequence.
[tex]\[
\frac{10}{5} = 2, \quad \frac{20}{10} = 2, \quad \frac{40}{20} = 2, \quad \frac{80}{40} = 2, \quad \frac{160}{80} = 2
\][/tex]
Each term is obtained by multiplying the previous term by 2, indicating this is a geometric sequence.
2. Determine the First Term and Common Ratio:
- The first term ([tex]\(a\)[/tex]) of the sequence is 5.
- The common ratio ([tex]\(r\)[/tex]) of the sequence is 2.
3. Write the General Formula for a Geometric Sequence:
The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is given by:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
where [tex]\(a\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the position of the term in the sequence.
4. Substitute the Known Values:
Substitute [tex]\(a = 5\)[/tex] and [tex]\(r = 2\)[/tex] into the formula:
[tex]\[
a_n = 5 \cdot 2^{n-1}
\][/tex]
5. Compare with the Provided Options:
Let's look at the provided options:
- A. [tex]\(a_n = 5(2)^{(n-1)}\)[/tex]
- B. [tex]\(a_n = 2(5)^{(n-1)}\)[/tex]
- C. [tex]\(a_n = 5(2)^n\)[/tex]
- D. [tex]\(a_n = 5 + 5(n-1)\)[/tex]
The derived formula [tex]\(a_n = 5 \cdot 2^{n-1}\)[/tex] exactly matches option A, [tex]\(a_n = 5(2)^{(n-1)}\)[/tex].
Hence, the explicit formula for the given sequence is:
[tex]\[
\boxed{a_n = 5(2)^{(n-1)}}
\][/tex]
