Lawn care isn't the right place for the "10th time is free" marketing. There are a few reasons for this.
For one, lawn care is not like other types of discretionary services. When they hire you, you are pretty much locked into a regular schedule. They don't hire you this week, Bob next week, and Steve the week after that. Businesses that would use the "free service after X number of purchases" are businesses that would be in direct regular competition for the job or sale week after week. Restaurants, car washes, things like that. A local sub shop used to give you a free 6" after you bought six subs. It works for businesses like that for the reason that it encourages faster and more frequent visits to earn the free sub. There is no faster and frequent option for you. They'll be back anyway. They're not going to be back more often than needed.
Second. People will get the idea that you are either desperate or worse, overcharging them by 10% to make up for the lost week. If they don't think this, they believe in things like free lunch. Not the customers you want.
Third. Rest assured there will be confusion. Little old Mrs. Smith will insist she has paid you nine times already after she has only paid you five times. It's hard enough to keep people straight with paying without this added piece of confusion. Imagine trying to keep track when you don't mow it at all for a few months.
And finally. What about Mrs Jones? The customer you already have. The one that gets wind of Mrs. Smith getting every tenth cut for free? You can explain why you are charging someone less. It's hard to explain why you are giving away free cuts to other customers.