Question : In a G.P. the sum of the first five terms is $80$, if the difference between the sixth and first term is $5$, find the first term and common ratio.
Ths question looks easy but I am not getting the answer, maybe I am making a mistake. Can I see how you will solve it so I can check where am I getting it wrong?
We have $a(1+r+r^2+r^3+r^4)=80$, and $ar^5-a=5$. We can factorise $r^5-1=(r-1)(1+r+r^2+r^3+r^4)$ then eliminate $r$ from the equations.
Using the above factorisation in the second equation we get $a(r-1)(1+r+r^2+r^3+r^4)=5$. Dividing this by the first equation gives $r=1.0625$.
To find $a$ substitute our value of $r$ back in, giving $a=\frac{5}{1.0625^5-1}$.