Consider the Recurrence Relation $$a _ { n } = 2 a _ { n - 1 } + 1$$

Consider the recurrence relation $$a _ { n } = 2 a _ { n - 1 } + 1$$ (a) Write the associated homogeneous recurrence relation.
(b) Find the general solution to the associated homogeneous recurrence relation.
(c) Find a particular solution to the given recurrence relation.
(d) Write the general solution to the given recurrence relation.
(e) Find the particular solution to the given recurrence relation when $$a _ { 0 } = 1$$