<s>[INST] 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?
 [/INST] Your expression is 
$$S = \sum_{k=1}^n k 10^k$$
We can pull out a factor of $10$ to get
$$S = 10 \sum_{k=1}^n k 10^{k-1}$$
Now, consider the function
$$f(x) = \sum_{k=1}^{n}kx^{k-1}$$
So $S = 10f(10)$. For $x \neq 1$, we can rewrite the sum as follows:
$$f(x) = \begin{align}
\frac{d}{dx}\sum_{k=1}^{n} x^k
&= \frac{d}{dx} \frac{x - x^{n+1}}{1-x}
\end{align}$$
where we have used the fact that $(x + x^2 + \cdots + x^n)(1-x) = x - x^{n+1}$ due to telescoping. 
Therefore the desired sum is
$$S = 10f(10) = 10 \left.\left(\frac{d}{dx} \frac{x - x^{n+1}}{1-x}\right)\right|_{x=10}$$
I'll let you carry out the differentiation to finish the problem.
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