To identify the 19th term of a geometric sequence where [tex]\( a_1 = 14 \)[/tex] and [tex]\( a_9 = 358.80 \)[/tex], we need to follow these steps:
1. Determine the common ratio [tex]\( r \)[/tex]:
We know that for a geometric sequence, the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Given [tex]\( a_9 = 358.80 \)[/tex]:
[tex]\[
a_9 = a_1 \cdot r^{(9-1)} \implies 358.80 = 14 \cdot r^8
\][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[
r^8 = \frac{358.80}{14} \implies r^8 = 25.6285714
\][/tex]
[tex]\[
r = \sqrt[8]{25.6285714} \implies r \approx 1.5
\][/tex]
2. Calculate the 19th term [tex]\( a_{19} \)[/tex]:
Now, with [tex]\( r \approx 1.5 \)[/tex], we determine [tex]\( a_{19} \)[/tex]:
[tex]\[
a_{19} = a_1 \cdot r^{(19-1)} \implies a_{19} = 14 \cdot (1.5)^{18}
\][/tex]
[tex]\[
a_{19} \approx 20689.88
\][/tex]
Hence, after rounding to the nearest hundredth, the 19th term of the geometric sequence, [tex]\( a_{19} \)[/tex], is [tex]\( \approx 20,689.88 \)[/tex].
The correct answer is:
[tex]\[ a_{19} \approx 20,689.88 \][/tex]
