Question

How do you write the formula for the nth term given 900, 300, 100, 33 1/3,...?

Answer 1

`900*(1/3)^(n-1)`

Explanation:

Each term after the first is equal to the previous term multiplied by `1/3`. If we do not simplify at each step, and simply group the `1/3`s together, then we can rewrite the sequence as

`900, 900*1/3, 900*(1/3)^2, 900*(1/3)^3, ...`

and from this it is evident that the `n^"th"` term may be written as `900*(1/3)^(n-1)`.

This type of sequence is called a geometric sequence. A geometric sequence is a sequence of the form `a_0, a_0r, a_0r^2, ...` where `a_0` is the starting term and `r` is the common ratio between terms. In this case, we'd have `a_0 = 900` and `r = 1/3`.