1. The irrational number $0.123456789101112 \ldots$ is formed by concatenating, in increasing order, all the positive integers. Find the sum of the first 2016 digits of this number after the decimal point.
Answer: 8499
Solution: Note that all the one-digit and two-digit numbers take up a total of $9+180=189$ digits. Thus, we stop at $\frac{2016-189}{3}+99=708$.
From 0 to 99 , there are 10 occurrences of 0 to 9 in the ones place, and ten iterations of 0 to 9 in the tens place. Thus, the total digit sum from 0 to 99 is $20(45)=900$. From 100 to 699 , there are 6 occurrences of 0 to 99, plus 100 iterations each of 0 to 6 (in the hundreds digit), which gives us a digit sum of $100(21)+6(900)$ $=7500$. This gives the numbers 700, 701, 702, 703, 704, 705, 706, 707, 708. These have a sum of 99 .
Combining everything we have, we obtain a total digit sum of $900+7500+99=8499$.