When setting up accounts on an iOS device, never use the "GMail" option -- it is worthless.
In fact, Google even tells you not to use it. I believe the phony support may be by design but that could be another article. Instead, use the "Microsoft Exchange" option just as Google tells you to do on their support website.
Here is a link to an article in the Google support archives that gives you all the details:
Setting up Google Sync with your iOS device
Follow these instructions and you will be able to sync everything perfectly.