Question

Given that………………………..

Answer 1

`\sum^(\infty)_(n=1) (a/b)^n=5 \\ \\ =(a/b)/(1-(a)/(b))=5 \\ \\ (a)/(b-a) =5 \\ \\ (a)/(b)=(5)/(6)`

So, we need to find

`\sum^(\infty)_(n=1) n(5/6)^n</p><p>`

Let this sum be S.

Then,

`S=(5/6)+2(5/6)^2 +3(5/6)^3+\cdots \\ \\ (5)/(6)S=(5/6)^2 + 2(5/6)^3+\cdots \\ \\ \implies (1)/(6)S=(5/6)+(5/6)^2+(5/6)^3+\cdots=5 \\ \\ \implies S=\boxed{30}`