Question: a write out the first few terms of the sequence...

a) Write out the first few terms of the sequence given by a₁=3; a_n=2a_n-1+4. Then find a recursive definition for 10, 24, 52, 108, ... .

b) Show that a_n=2ⁿ-5ⁿ is a solution to the recursion a_n=7a_n-1-10a_n-2. What would the initial conditions need to be for this to be the closed formula for the sequence?

c) Use summation () notion to rewrite 1+5+9+13+...+425.

d) Find a closed formula for the sequence with recursive definition a_n=2a_n-1-a_n-2 with a₁=1 and a₂=2.

e) Suppose you draw n lines in the plane. What is the maximum number of regions (call the regions R_n) that can be created, including unbounded regions? Find a recursive formula for the number of regions created by the n lines and justify your answer. Also guess at a closed formula based on your data table.