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?

Put
$$ S_n = \sum_{k=1}^{n} k\cdot 10^k.$$
Then:
$$ 9 S_n = (10-1)S_n = \sum_{k=1}^{n}k\cdot 10^{k+1}-\sum_{k=1}^{n}k\cdot 10^k=\sum_{k=2}^{n+1}(k-1)\cdot 10^k-\sum_{k=1}^{n}k\cdot 10^k$$
hence:
$$ 9 S_n = n\cdot 10^n -\sum_{k=1}^n 10^k = n\cdot 10^n-\frac{10^{n+1}-10}{9},$$
so:
$$ S_n = \color{red}{\frac{10}{81}\left(1+(9n-1)\cdot 10^n\right)}.$$
