Question

`s = (a(r^n -1))/(r-1)` Making `'r'` the subject formula..?

Answer 1

This is not generally possible...

Explanation:

Given:

`s = (a(r^n-1))/(r-1)`

Ideally we want to derive a formula like:

`r = "some expression in " s, n, a`

This is not going to be possible for all values of `n`. For example, when `n=1` we have:

`s = (a(r^color(blue)(1)-1))/(r-1) = a`

Then `r` can take any value apart from `1`.

Also, note that if `a=0` then `s=0` and again `r` can take any value apart from `1`.

Let us see how far we can get in general:

First multiply both sides of the given equation by `(r-1)` to get:

`s(r-1) = a(r^n-1)`

Multiplying out both sides, this becomes:

`sr-s=ar^n-a`

Then subtracting the left hand side from both sides, we get:

`0 = ar^n-sr+(s-a)`

Assuming `a!=0`, we can divide this through by `a` to get the monic polynomial equation:

`r^n-s/a r+(s/a-1) = 0`

Note that for any values of `a, s` and `n` one root of this polynomial is `r=1`, but that is an excluded value.

Let us attempt to factor out `(r-1)`...

`0 = r^n-s/a r+(s/a-1)`

`color(white)(0) = r^n-1-s/a(r-1)`

`color(white)(0) = (r-1)(r^(n-1)+r^(n-2)+...+1-s/a)`

So dividing by `(r-1)` we get:

`r^(n-1)+r^(n-2)+...+1-s/a = 0`

The solutions of this will take very different forms for different values of `n`. By the time `n >= 6`, it is not generally solvable by radicals.