I need to find the summation of $ab^{-k}$ from $k=5$ to $n$ using Gauss' Law. Here's what I have so far:
$$\begin{align}S_n&=(ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}+ab^{-n}+ab^{-(n-1) }+ab^{-(n-2) } +\cdots+ab^{-5})\\2S&=(\frac{a}{b^5} +\frac{a}{b^n})+(\frac{a}{b^6}+\frac a{b^{-n+1}} )+(\frac a{b^n} +\frac{a}{b^{-n+2}} )+\cdots+(\frac{a}{b^n} +\frac{a}{b^5} ) \\ 2S&=(\frac a{b^5}) (1+\frac 1 b+\frac 1 {b^2} +\cdots+\frac{1}{b^{n-4}})(n-4)\end{align}$$
From here I'm confused how to get the answer. What should my next step be?
$$ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}=ab^{-n}\left(1+b+b^2+\cdots+b^{n-5}\right)$$ $$ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}=\dfrac{ab^{-n}}{b-1}\left(b^{n-4}-1\right),\,\,\,\,\,b\not=1.$$ $$ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}=\dfrac{a}{b-1}\left(\dfrac{1}{b^4}-\dfrac{1}{b^n}\right)$$