Question

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

−312,500
−12,500
62,500
1,562,500

Answer 1

First, we are going to find the common ratio of our geometric sequence using the formula:
`r= (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= (-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