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:
- Set \(a_1 = 1600\), \(r = \frac{3}{4}\), and \(n = 5\):
\[ a_5 = 1600 \cdot \left(\frac{3}{4}\right)^{(5-1)} \]
- Calculate \( \left(\frac{3}{4}\right)^{4} \):
\[ \left(\frac{3}{4}\right)^{4} = \frac{3^4}{4^4} = \frac{81}{256} \]
- Now substitute that back into the equation:
\[ a_5 = 1600 \cdot \frac{81}{256} \]
- To simplify, first calculate \(1600 \div 256\):
\[ 1600 \div 256 = 6.25 \]
- Now multiply by 81:
\[ a_5 = 6.25 \cdot 81 = 506.25 \]
Thus, the value of \(a_5\) for the geometric sequence is:
\[ \boxed{506.25} \]