Michigan Rummy? Incorrect Name?
This is a form of contract rummy called progressive rummy.
(I note that since the contracts you name include ever-increasing ones eventually leading to ones that cannot be met with only seven cards that each deal probably involves an additional card)

Progressive Rummy: Progressive Rummy is played very similarly to Contract Rummy. The main differences are in the number of cards dealt in each of the seven deals. The first deal starts with 6 cards dealt, the second deal 7 cards are dealt, and so on until the 7th deal where 12 cards are dealt. The initial meld requirements are the same as in standard contact rummy. The player generally completes the hand by playing all his cards to the table to meet the contract, as he will usually not have any additional cards in his hand (unless he opted, on another player's turn to take the current upcard, as described in the rules for Contract Rummy, above). If this is the case, the player may play cards to other melds on the table (if any) once they have made their initial meld. A player may also play their last card by discard, at which time they also win the deal. Scoring in progressive rummy is the same as in Contract Rummy, except that jokers are worth 25 points each.