1.
Explain what the linearity of (the solutions of) a recurrence relation
means, and prove it for the homogeneous firstorder linear recurrence
relations.
The set of solutions of a linear recurrence relation forms a linear space.
That is, if
a
n
=
f
(
n
) and
a
n
=
g
(
n
) are two solutions for such a
recurrence, then
a
n
=
f
(
n
) +
g
(
n
) and
a
n
=
cf
(
n
) (for an arbitrary
constant
c
) are also solutions.
A homogeneous firstorder linear recurrence relation is a relation of
form
a
n
+1
=
k
(
n
)
a
n
(often, we deal with the case that
k
(
n
) is a con
stant).
If
a
n
=
f
(
n
) and
a
n
=
g
(
n
) satisfy this recurrence, then
f
(
n
+ 1) +
g
(
n
+ 1) =
k
(
n
)
f
(
n
) +
k
(
n
)
g
(
n
) =
k
(
n
)(
f
(
n
) +
g
(
n
)),
thus
a
n
=
f
(
n
) +
g
(
n
) also satisfies it.
Similarly, we can prove that
a
n
=
cf
(
n
) is also a solution.
2.
Chapter 10.1, exercises 2 (in the cases (a) and (b), the solution is not
unique, despite what the statement of the exercise suggests. Describe
all the solutions, instead) and 6.
10.1.2 a)
a
n
=
c
1
.
5
n
b)
a
n
=
c
parenleftBig
5
4
parenrightBig
n
c)
a
n
=
15
4
parenleftBig
4
3
parenrightBig
n
d)
a
n
= 16
parenleftBig
3
2
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 Spring '09
 MarniMishna
 Characteristic polynomial, Recurrence relation, Fibonacci number, firstorder linear recurrence, linear recurrence relation

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