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.

  1. 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.

  2. 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 \(n \ge 1\text{.}\)

  3. Find a closed formula for the \(n\)th term of the sequence and prove it is correct by induction.

in-context