###### Investigate!

Each day your supply of magic chocolate covered espresso beans doubles (each one splits in half), but then you eat 5 of them. You have 10 at the start of day 0.

- Write out the first few terms of the sequence. Then give a recursive definition for the sequence and explain how you know it is correct.
- Prove, using induction, that the last digit of the number of beans you have on the \(n\)th day is always a 5 for all \(n \ge 1\text{.}\)
- Find a closed formula for the \(n\)th term of the sequence and prove it is correct by induction.