52. (10-11 grades) Find the sum $1+2 a+3 a^{2}+4 a^{3}+\ldots+n a^{n-1}$
52. Multiply each term of the sum by $a$. Denoting the desired sum by $S$, we will compute the difference $S-S a$, then compute $S$

$$
S=\frac{1-a^{n}-n a^{n}+n a^{n+1}}{(1-a)^{2}}
$$