As I said in the (now deleted) comment, I don't think that is really a "bash" solution.

In pure bash you can do something like this, looping once:

 #!/bin/bash
 
 COUNTER=0
 TOTALSUM=0
 A=0
 while [ $COUNTER -lt 1000 ]; do
 let A=COUNTER%5
 if [[ $A = 0 ]]; then
 let TOTALSUM=TOTALSUM+COUNTER
 else
 let A=COUNTER%3
 if [[ $A = 0 ]]; then
 let TOTALSUM=TOTALSUM+COUNTER
 fi
 fi
 let COUNTER=COUNTER+1 
 done
 
 echo $TOTALSUM

But if you want the "one-liner" version (which is not really one line after all):

 for ((i = 0; i < 1000; i += 1)); do let A=i%5; let B=i%3; if [[ $A = 0 || $B = 0 ]]; then let TOTALSUM=TOTALSUM+i; fi; done; echo $TOTALSUM