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

###### Question details

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

b) Show that a_{n}=2^{n}-5^{n} 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
a_{1}=1 and a_{2}=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.