The support for a deck is from the post under neath it. Each 5 gallons of water weighs about 65lbs. If I was doing your job you would get 2 more rows of posts and beams underneath the space where the tub is. What makes a deck sag is the weight applied to the footing pad under the post. The greater the area of the footing pad the more supprt you will have. you have to think of it as a single post with 5,000 pounds on it is like a needle into the ground. There is no mass to stop the penetration of the post into the ground. The more post, with footings you have applied over an area the less sinking into the grouind you will have. Generaly a 12" diameter pad 10" thick will do the job for each individual post.
To see what happens to improperly supported decks go to www.deckfailures.com