TITLE: A "fast" way to find the sum of the sequence $5,5.5,5.55,5.555,5.5555,\ldots $ (20 terms)
QUESTION [6 upvotes]: My initial approach is diving the whole sum by $9$ and taking the common $5$ out which gives $$\frac{5}{9}[(10-1)+(10-0.1)+(10-0.01)+\cdots + (10-10^{-19})]$$ after some algebra this could be reduced to $$\frac{5}{9} \times [200-\frac{10}{9} \times (1-10^{-20})]$$
after this I am not sure how to show that is almost equal to $110.5$? Also if any body wants to suggest any other tricky/fast way I will appreciate it.

REPLY [7 votes]: $5+5.5+5.55+5.555+\cdots $
$= 5 + (5+0.5) + (5+0.5+0.05) + \cdots$
$= 20 \times 5 + 19 \times 0.5 + 18 \times 0.05 + 17 \times 0.005 \cdots + 1 \times 0.00{\ldots}05$
$\approx 100+9.5+0.9+0.085 $
$= 110.485$.