The first term of a geometric series is 27, the last term is 8 and the sum of the series is 65. What is the common ratio and how many terms are there in the series?
$\displaystyle a_1=27$
$\displaystyle a_n=8\Rightarrow a_1q^{n-1}=8\Rightarrow q^{n-1}=\frac{8}{27}$
$\displaystyle S_n=65\Rightarrow a_1\frac{q^n-1}{q-1}=65\Rightarrow\frac{q^n-1}{q-1}=\frac{65}{27}$
$\displaystyle q^n=q^{n-1}\cdot q=\frac{8}{27}q$
Then, $\displaystyle \frac{\frac{8}{27}q-1}{q-1}=\frac{65}{27}\Rightarrow q=\frac{2}{3}$
$\displaystyle \left(\frac{2}{3}\right)^{n-1}=\frac{8}{27}\Rightarrow n=4$