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Question
Calculate lower a subscript 5 baseline for the geometric sequence in which lower a subscript 1 baseline equals 1600 and the common ratio is three-fourths.
(1 point)
Responses

506.25
506.25

379.6875
379.6875

675
675

1,200

Answer 1

In a geometric sequence, the \(n\)-th term can be calculated using the formula:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term (1600 in this case),
- \(r\) is the common ratio (\(\frac{3}{4}\) in this case),
- \(n\) is the term number.

To find \(a_5\), we substitute the values into the formula:

1. Set \(a_1 = 1600\), \(r = \frac{3}{4}\), and \(n = 5\):

\[
a_5 = 1600 \cdot \left(\frac{3}{4}\right)^{(5-1)}
\]

2. Calculate \( \left(\frac{3}{4}\right)^{4} \):

\[
\left(\frac{3}{4}\right)^{4} = \frac{3^4}{4^4} = \frac{81}{256}
\]

3. Now substitute that back into the equation:

\[
a_5 = 1600 \cdot \frac{81}{256}
\]

4. To simplify, first calculate \(1600 \div 256\):

\[
1600 \div 256 = 6.25
\]

5. Now multiply by 81:

\[
a_5 = 6.25 \cdot 81 = 506.25
\]

Thus, the value of \(a_5\) for the geometric sequence is:

\[
\boxed{506.25}
\]