The range is 1 to 2*500-1=999.
Those divisible by 3 form a series, 3, 9, 15, 21, ... these being 3*1, 3*3, 3*5, 3*7, ...3*333. The general term is 3(2n-1) and the last term is 3*333 so 2n-1=333, 2n=334, n=167.
Those divisible by 5 form the series: 5, 15, 25, ...995, or 5*1, 5*3, 5*5, ...5*199. The general term is 5(2n-1) and 2n-1=199 for the last term, so n=100.
The number of multiples of 3=167 and 5=100. But some are divisible by both 3 and 5 (i.e., 15).
The series for 15 is 15, 45, 75, ...975, or 15(2n-1) and 2n-1=65, n=33.
If we now add the totals for 3 and 5, we get 267 but we need to subtract 33 from this so as to avoid double accounting for numbers divisible by both 3 and 5. 267-33=234. There are 234 odd numbers divisible by 3 or 5 between 1 and 999, which is the range of the first 500 odd numbers. This includes numbers divisible by both. The number divisible by 3 or 5 but not both is 234-33=201.