Question:

I attempted to answer it.

The problem I have is with the worked solutions is how did the author come up with:

$\displaystyle \frac{1}{a}(\frac{r^{n-1}+r^{n-1}+r^{n-2}+r^{n-31}+...+r^{0}}{r^{(n-1)(n-2)(n-3)..3.2.1}}) $

I thought that $\displaystyle \frac{1}{a}+\frac{1}{ar} + ....+ \frac{1}{ar^{n-1}} = \frac{1}{a}(r^{0}+r^{-1}+r^{-2}+ r^{-3}) $

therefore r = common ratio = r^{-1}

and lastly how did the author need up with $\displaystyle \frac{a^{n-1}n(n-1)}{2P_{n}} $ where $\displaystyle P_{n} = a^{n}\frac{r^{n(n-1)/2}}{2} $ as the answer

I thought $\displaystyle P_{n} = a^{n}\frac{r^{n(n-1)/2}}{2} $ is wrong.

Shouldn't $\displaystyle P_{n} = a^{n}r^{\frac{n(n-1)}{2}} $ ?

Please if someone can be kind to explain to me thank you

Edit: Please don't worry about the second part of the question

$\displaystyle \frac{1}{(a)(ar).....(ar^{n-1})} $ I did this question