Note:
I redid this problem. For some strange reason I didn't notice that it said "odd
numbers" and I did it for all the numbers. I don't know why I didn't see it,
It says "odd" plain as day! Anyway, my bad. It's the same principle, only a
little harder. Here's my approach:
2-digit numbers:
The sequence of odd numbers:
51, 53, 55, ..., 97, 99
have 2 digits each. There are the same number of those as there are
numbers which are 1 less than them, which are these:
50, 52, 54, ..., 96, 98,
There are the same number of those as there are
numbers which are half of them, which are these:
25, 26, 27, ..., 48, 49
These are the 49 numbers 1 through 49, minus the 24 numbers 1 through 24
and 49-24 = 25
So there are 25 numbers with 2 digits each, so the number of digits in
the 2-digit numbers in the original sequence is 2x25 or 50.
---------------------------------------------
3-digit numbers:
The sequence of odd numbers:
101, 103, 105, ..., 997, 999
have 3 digits each. There are the same number of those as there are
numbers which are 1 less than them, which are these:
100, 102, 104, ..., 996, 998,
There are the same number of those as there are
numbers which are half of them, which are these:
50, 51, 52, ..., 498, 499
These are the 499 numbers 1 through 499, minus the 49 numbers 1 through 49
and 499-49 = 450
So there are 450 numbers with 3 digits each, so the number of digits in
the 3-digit numbers in the original sequence is 3x450 or 1350.
---------------------------------------------
4-digit numbers:
The sequence of odd numbers:
1001, 1003, 1005, ..., 4999, 5001
have 4 digits each. There are the same number of those as there are
numbers which are 1 less than them, which are these:
1000, 1002, 1004, ..., 4998, 5000,
There are the same number of those as there are
numbers which are half of them, which are these:
500, 501, 502, ..., 2499, 2500
These are the 2500 numbers 1 through 2500, minus the 499 numbers 1 through
499
and 2500-499 = 2001
So there are 2001 numbers with 4 digits each, so the number of digits in
the 4-digit numbers in the original sequence is 4x2001 or 8004.
---------------------------------------------
Grand total: 50 + 1350 + 8004 = 9404 digits.
Edwin
