Question

How do you write an equation for the nth term of the geometric sequence `4, 8, 16, ...`?

Answer 1

`a_n=4 * 2^(n-1)`
...but see below

Explanation:

It seems that currently the most common usage is that the initial value of a sequence is considered the "first" value.
That is for the given geometric sequence: 4, 8, 16, ...
`color(white)("XXX")a_1=4`
Each term after the first increases the immediately prior term by a factor of `2` and since there are `(n-1)` terms after the first for the `n^(th)` term, we have
`color(white)("XXX")a_n = a_1 xx underbrace(2 xx 2 xx ... xx2)_(n" times") = a_1 * 2^(n-1) = 4 * 2^(n-1)`

Less common these days (but the version I learned long ago) was to call the initial value of the sequence, the `"zero"^(th)` value.
That is for the given sequence: 4, 8, 16, ...
`color(white)("XXX")a_0=4`
and the equation for the `n^(th)` value would be
`color(white)("XXX")a_n=a_0 * 2^n` or `4 * 2^n`