Question

How do you find a formula for the nth term of the geometric sequence: 500, 100, 20, 4, ...?

Answer 1

`t_n=a(1/5)^(n-1)`

Explanation:

Recall that the formula for the `n^("th")` of a geometric sequence is:

`color(blue)(|bar(ul(color(white)(a/a)t_n=ar^(n-1)color(white)(a/a)|)))`

where:
`t_n=`term number
`a=`first term
`r=`common ratio
`n=`number of terms

Start by determining the value of `a`, the first term of the sequence. In your case, that would be `500`.

Use `a=500`, and `t_2=100` to create an equation representing the second term.

`t_n=ar^(n-1)`

`100=500r^(2-1)`

Solve for `r`.

`r=1/5`

Now that you have the value of `r`, you can make a formula representing the `n^("th")` term of the geometric sequence.

`color(green)(|bar(ul(color(white)(a/a)t_n=a(1/5)^(n-1)color(white)(a/a)|)))`