Evaluating the sum $1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + \dots + n\cdot 10^n$ How can I calculate 
$$1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + 4\cdot 10^4+\dots  + n\cdot 10^n$$
as a expression, with a proof so I could actually understand it if possible?

Let $$S_n = 1*10^1+2*10^2+3*10^3 + \cdots n*10^n$$
$$10S_n = 1*10^2+2*10^3+3*10^4  + \cdots n*10^{n+1}$$
$$S_n - 10S_n = 1*10^1 + 1*10^2 + 1*10^3 +\cdots1*10^n-n*10^{n+1}$$
$$-9S_n = 10*\frac{1-10^n}{1-10} - n*10^{n+1}$$
Simplifying, $$S = \frac{10}{81}*(9*10^n n-10^n+1)$$
