<s>[INST] Expression generating $\left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right)$ I'm looking for a closed-form expression (in terms of $n$), that will give the sequence
$$
(s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right).
$$
Can anyone think of one? I made a related post to this question several minutes ago but I realized I was interpreting the sequence wrong.
 [/INST] $$s_n=\sum_{k=1}^n\frac{3\sum_{l=0}^{k-1}{10^l}}{10^k}.$$ Using the geometric sequence sum formula this simplifies considerably to:
$$s_n=\frac{1}{27} (9n-1 + 10^{-n} ). $$
</s>