Series: Method of Difference

Given S = (4/5) + (7/5^2) + (10/5^3) + ... + (31/5^10), express S in Σ form.

By considering the difference between S and (1/5)S, find S.

So this is how I did it:

S = Σ (3r+1)/5^r (Upperlimit 10, Lower limit r = 1 )

(1/5)S = Σ (3r+1)/5^(r+1)

S - (1/5)S = Σ [(3r+1)/5^r - (3r+1)/5^(r+1)]

= + 5/4 - 4/25 r = 1

+ 1/25 - 7/125 r = 2

+ 10/125 - 10/625 r = 3

+ ...

I use the method of difference to solve but there don't seem to be any terms I can cancel away... Any idea what went wrong? (Thinking)

THanks in advance!!