You are hurting yourself if you give people a discount who decide to pay monthly because you are giving them savings (lowering the money that YOU will be getting) with absolutely no reason. If the reason that you want contracts is to get people to pay monthly instead of weekly then just inform your clients that they will be billed monthly for maintenance services and per job for other things such as mulching if you wanted. Or set it up however you want but you don't need to provide an incentive for clients based on how you bill(unless they are prepaying). Great part about being a business owner is you make the rules.
I would say this though, its not a bad thing that people are paying weekly. Cash flow is huge. Most larger businesses do monthly because thats how most any bill you will ever come across is structured so its considered the norm. Credit cards, trash pickup, mortgage, insurance etc are all monthly bills so people are used to that. I wish all our customers paid immediately upon service but that's not realistic
What are you planning on going to school for? Do you want to be in the landscape industry after college? How much did you guys gross this year? 20-30k?