Backups twice as large from one day to the next

My SQL Server 2008 R2 production database is running on a dedicated Windows server. It has been running for more than 20 years. In the last period we have suffered some problems such as freezing without memory but with more than 100 GB RAM available according to the task manager. In addition, the auto increment indexes are skipped, example of 720 jumps to 723. Every Sunday has an automatic task to rebuild indexes. Last time, the backup went from 22 GB to 37 GB. Between finishing the task for the indexes and completing backup took about 12 hours. Since then, it has grown 1.5 GB in less than 12 hours. I read a bit about rebuilding indexes could be the cause and also that the growth is not in % if not that it grows 1GB above what is needed. An important point is that the tables and data are the same size; that is to say the data does not grow. Could you help me to find the reason for this.