The total number of pages from page X to page Y is Y minus X plus 1.
So from 1 to 9, there are 9 minus 1 is 8 plus 1 is 9.

Then, in summary:

There are:
   9 pages numbered      1 through     9 that need 1 digit each:     9 times 1
  90 pages numbered     10 through    99 that need 2 digits each:   90 times 2
 900 pages numbered    100 through   999 that need 3 digits each:  900 times 3
9000 pages numbered   1000 through  9999 that need 4 digits each: 9000 times 4
1522 pages numbered  10000 through 11521 that need 5 digits each: 1522 times 5 

Do the arithmetic.