Question

What is the 9th term of the geometric sequence 4, −20, 100, …?

Answer 1

First, we are going to find the common ratio of our geometric sequence using the formula: $r= \frac{a_{n}}{a_{n-1}}$. For our sequence, we can infer that $a_{n}=-20$ and $a_{n-1}=4$. So lets replace those values in our formula:
$r= \frac{-20}{4}$
$r=-5$

Now that we have the common ratio, lets find the  explicit formula of our sequence. To do that we are going to use the formula: $a_{n}=a_{1}*r^{n-1}$. We know that $a_{1}=4$; we also know for our previous calculation that $r=-5$. So lets replace those values in our formula:
$a_{n}=4*(-5)^{n-1}$

Finally, to find the 9th therm in our sequence, we just need to replace $n$ with 9 in our explicit formula:
$a_{9}=4*(-5)^{9-1}$
$a_{9}=4*(-5)^{8}$
$a_{9}=1562500$

We can conclude that the 9th term in our geometric sequence is 1,562,500