Question

What is the $1000^{\rm th}$ positive integer with an odd number of digits?

Answer 1

10090

Explanation:

There are 9 one-digit positive integers (1 through 9). There are 900 three-digit positive integers, since there are 100 three-digit numbers with each hundreds digit from 1 to 9. So far, we've counted 909 positive integers with an odd number of digits. Next are five-digit numbers. Since 909 + 91=1000, we're looking for the 91st five-digit number.

The first five-digit number is 10000, so the 91st five-digit number is 10000 + 90, or 10090.

Answer 2

9 numbers have one digit (1-9)
900 have three digits (100-999)
90000 have five digits (10000-99999)

Clearly the 1000th positive integer with an odd number of digits will have five digits. Since there are 909 integers with up to three digits, the 91st integer in the third range will be the one you want. That number is 10090.