Question

The positive integers $1,$ $2,$ $3,$ $\dots$ are listed in order. What is the $1000^{\rm th}$ positive integer with an odd number of digits

Answer 1

There are 9 positive integers with 1 digit (1 - 9).

There are 90 integers with 2 digits (10 - 99).

There are 900 integers with 3 digits (100 - 999).

There are 9000 integers with 4 digits (1000 - 9999).

And so on. It stands to reason that there are $9\times10^{n-1}$ integers with $n$ digits.

Now,

1000 = 9 + 900 + 91

so the 1000th positive integer with an odd number of digits is the 91th number with 3 digits, which is 190.