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

To find the 1000th positive integer with an odd number of digits, we need to determine the range of positive integers that satisfy this condition. The 1st to 9th positive integers have 1 digit, the 10th to 99th have 2 digits, and so on. The 1000th positive integer with an odd number of digits is 1000.

Step-by-step explanation:

To find the $1000^{
m th}$ positive integer with an odd number of digits, we need to determine the range of positive integers that satisfy this condition. Let's consider the first few positive integers:

1. $1$ (1 digit)
2. $2$ (1 digit)
3. $3$ (1 digit)
4. $4$ (1 digit)
5. $5$ (1 digit)
6. $6$ (1 digit)
7. $7$ (1 digit)
8. $8$ (1 digit)
9. $9$ (1 digit)
10. $10$ (2 digits)
11. $11$ (2 digits)
12. $12$ (2 digits)
13. $13$ (2 digits)
14. $14$ (2 digits)
15. $15$ (2 digits)
16. $16$ (2 digits)
17. $17$ (2 digits)
18. $18$ (2 digits)
19. $19$ (2 digits)
20. $20$ (2 digits)

From this pattern, we can observe that the $1^{
m st}$ to $9^{
m th}$ positive integers have 1 digit, the $10^{
m th}$ to $99^{
m th}$ positive integers have 2 digits, and so on.

To find the range of positive integers with 3 digits, we need to determine the number of integers in the range from $100$ to $999$.

This range contains $999 - 100 + 1 = 900$ integers. So, the $1000^{
m th}$ positive integer with an odd number of digits is the first positive integer with 4 digits, which is $1000$.