`(( ))` is an arithmetic evaluation block. Its content has to be an arithmetic expression. Arithmetic expressions don't require `$` before simple variable names and some more complex expressions like array indexing.

`(( i + 1 ))` will evaluate the result of adding one to `$i` and then throw away the result. It doesn't do anything useful (other than having a different exit code depending on whether the expression evaluated to 0 or not).

`$(( ... ))` evaluates the expression and then returns its value. ie `i=$(( i + 1 ))` will increment `$i`, just like `(( i++ ))` did.

To add arbitrary numbers, you'd have to do an assignment, like this:

  [mmh@x670]$ ((i=i+2)); echo $i
  6
  [mmh@x670]$ ((i=i+2)); echo $i
  8