Consider the statement ``If $a_n = 5^n - 2^n$, then $3 | a_n$ for all integers $n>0$.''
  1. List the first 4 terms in this sequence and show that they are all divisible by $3$.
  2. First, do a proof in a manner similar to that we used in class. Use the proofs in the textbook as examples of how to format and present things.
  3. Do a second proof in two steps. A) Prove that this sequence of integers satisfies the recurrence equation $a_{n+2} - 7 a_{n+1} + 10 a_n = 0$. B) Argue by induction based on this recurrence equation. (Be careful about the initial conditions. You'll need to know both $3|a_1$ and $3|a_2$.)
  4. Show that any sequence of the form $b_n = c_1 5^n + c_2 2^n$ for any integers $c_1$ and $c_2$ solves the recurrence equation \[x_{n+2} - 7 x_{n+1} + 10 x_n = 0.\]
  5. Find a choice of $c_1$ and $c_2$ such that all the terms in the sequence are divisible by $3$.
  6. Find a choice of $c_1$ and $c_2$ such that some terms in the sequence are NOT divisible by $3$.