All the positive triangular numbers are written successively without commas or spaces resulting in this infinite string.
13610152128364555667891105120136153171..............
Determine the 2010^{th} digit in the above pattern.
1. The formula for the nth triangular number is 1/2n(n+1).
2. for small n it's probably easiest to set out the terms:
 3 less than 10, for 3 digits
 another 10 less than 100, for 20 digits
 another 31 less than 1000, for 93 digits
3. Then 1/2n(n+1) = 10000 gives n about 141, and n = 141 is in fact 10011. So we get n = 45 to 140 inclusive for an additional 96 numbers of 4 digits each: 384 digits.
4. Using the same method, 1/2n(n+1) = 100000 gives n about 446, and n=447 is in fact 100128. So we get n = 141 to 446 inclusive for an additional 306 numbers of 5 digits each: 1530 digits and the running total is over 2010; to be exact, it is 2030.
5. So the sought digit is the last digit of the fourth number before n = 446, namely n = 442, which is 97903: the last digit of which is a 3.

Posted by broll
on 20100720 12:46:36 