I think most have misunderstood the original post, or maybe I did but the OP is stating first 1 time cut for 10$.
I don't know your location but example here is we average 25 mowings a season. Your basic yard costs $35. 25x$35=875. minus 1 free $35= $840 / by 25 mowings New price is $33.60. I have lost $1.40 per week on the cut. Not much money to gain a customer. Most spend considerably more for customer acquisition.
Just to really get you going we ran an ad for a 2 week period this spring for 1 month of free mowing. If you are just giving the free or discounted mowing away and not signing some type of term contract, then yes you just lost money. We gave away your last month of mowing free with the 1 year commitment. Most then signed up for 3-4 other services that have higher profit margins. (this was the point) For the deal to be valid you had to stay committed all season. As well as the last month of mowing for us is pretty easy with very little lawn growth from week to week. This has it being or lowest overhead month. I can assure you we did well with every customer we signed up this way and made money on these home along with increasing profits on existing with tighter routes.