It's Not Magic

\[\dfrac{19}{81}\approx 0.\red{2}\blue{3}\red{4}\blue{5}\red{6}\blue{7}\red{9}\red{0}\blue{1}...\] \[\dfrac{199}{9801} \approx 0.\red{02}\blue{03}\red{04}\blue{05}\red{06}\blue{07}\red{08}\blue{09}\red{10}\blue{11}...\] \[How\space to\space generate\space such\space rationals?\] \[let \space there \space be \space a \space sequence \space a_n\] \[a_1=\alpha\beta\] \[\forall n\in\mathbb{N},a_{n+1}=(\alpha+nd)\beta r^n\] \[let \space S_n = \sum_{k=1}^{n}a_k\] \[=\sum_{k=1}^{n} (\alpha+(k-1)d)\beta r^{k-1}\] \[=\sum_{k=1}^{n} (\alpha\beta r^{k-1}+(k-1)d\beta r^{k-1})\] \[=\alpha\beta\sum_{k=1}^{n}r^{k-1}+\beta d\sum_{k=1}^{n}(k-1)r^{k-1}\] \[=\alpha\beta\dfrac{r^{n}-1}{r-1}-\beta d\sum_{k=1}^{n}r^{k-1}+\beta d\blue{\sum_{k=1}^{n}kr^{k-1}}\] \[=\beta\dfrac{r^{n}-1}{r-1}(\alpha-d)+\beta d\blue{S}\] \[\blue{S} =\sum_{k=1}^{n}kr^{k-1}\] \[S\times r=\sum_{k=1}^{n}kr^{k}=nr^n+\sum_{k=2}^{n}(k-1)r^{k-1}\] \[\Rightarrow S(1-r)=-nr^n+\sum_{k=1}^{n-1}r^k\] \[=\dfrac{r^n-1}{r-1}-nr^n\] \[\Rightarrow S=\dfrac{r^n-1}{(r-1)^2}-\dfrac{nr^n}{r-1}\] \[Putting \space this\space we\space get\] \[S_n=\dfrac{\beta(1-r^n)(\alpha(1-r)+dr)}{(1-r)^2}-\dfrac{\beta dnr^n}{1-r}\] \[\Rightarrow \sum_{k=1}^{\infty}a_k=\dfrac{\beta(\alpha(1-r)+dr)}{(1-r)^2}\] \[Put\space \alpha=1=\beta=d, r=10^{-n},n\in\mathbb{N}\] \[\Rightarrow \sum_{k=1}^{\infty}a_k=\dfrac{100^n}{(10^n-1)^2}\] \[Now\space consider B_n=\dfrac{100^n}{(10^n-1)^2}-1\] \[\Rightarrow \boxed{B_n=\dfrac{2\times 10^n - 1}{(10^n - 1)^2}}\] \[Which\space is \space the \space Formula \space we \space wanted!\]