Answer
The ninth term .
Work Step by Step
A geometric sequence has the form$ a, ar, ar^{2}, ar^{3}, \ldots$
The nth term of the sequence is $\quad a_{n}=ar^{n-1}$
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$a_{2}=30,\qquad a_{5}=3750$
$\displaystyle \frac{a_{5}}{a_{2}}=\frac{3750}{30}=125$
Also, $\displaystyle \frac{a_{5}}{a_{2}}=\frac{ar^{4}}{ar^{1}}=r^{3}$
So $r^{3}=125 \Rightarrow r=5$
From $a_{3}=ar^{2}\displaystyle \Rightarrow\quad a=\frac{a_{3}}{r^{2}}=\frac{30}{25}=\frac{6}{5}$
Finally, from $a_{n}=ar^{n-1}=\displaystyle \frac{6}{5}\times 5^{n-1}=468,750$
we find n:
$\displaystyle \frac{6}{5}\times 5^{n-1}=468,750\qquad/\times\frac{5}{6}$
$5^{n-1}=390625$
Factoring, we find $390625=25^{4}=5^{8}$
$5^{n-1}=5^{8}$
$n-1=8$
$n=9$
The ninth term is $468,750$.
