Integration — practice question

A television quiz show is broadcast every day. On day 1 the prize money is $\$1000$. If nobody wins it, the prize fund is raised for day 2. This pattern of increase continues each day until someone wins. The television company considered the following two different models for raising the prize money. Model 1: Increase the prize money by $\$1000$ each day. Model 2: Increase the prize money by $10\%$ each day. On each day that the prize money is not won, the television company makes a donation to charity. The donation is $5\%$ of the prize on that day. After $40$ days the prize money has still not been won.