A 100 Streak Goal

Probability Level 2

Daniel wants to get a 100-day streak on Brilliant.org. He plans to do so in the following manner:

On the first day, he does $1$ problem. On the second day, he does $2$ problems. On the third day, he does $3$ problems. This pattern continues on until the $10$ th day. On the $10$ th day, he does $1+0=1$ problem. On the $11$ th day, he does $1+1=2$ problems. In general, on the $\overline{ab}$ th day, he does $a+b$ problems. Finally, on the $100$ th day, he does $1+0+0=1$ problem, completing his streak.

In total, how many problems did he do?