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Fall 2009
CS131 – Combinatorial Structures
Homework 8
Homework 8, due Nov 17
You must prove your answer to every question.
Do not rely only on the homework for exercise: there are several selfcheck ex
ercises of the easier kind in the book, try to solve them, too!
Problem 1
(LPV 3.8.4)
.
(10pts) Find all values of
n
and
k
for which
(
n
k
+
1
)
=
3
(
n
k
)
.
Solution.
We have
±
n
k
+
1
!
=
n
(
n

1)
···
(
n

k
+
1)(
n

k
)
k
!(
k
+
1)
,
±
n
k
!
=
n
(
n

1)
···
(
n

k
+
1)
k
!
.
The equality is therefore
n
(
n

1)
···
(
n

k
+
1)(
n

k
)
k
!(
k
+
1)
=
3
n
(
n

1)
···
(
n

k
+
1)
k
!
,
n

k
k
+
1
=
3.
Rearrangement gives
n
=
4
k
+
3. The equation holds for
k
=
0,1,2,.
.., with
n
=
4
k
+
3.
Problem 2.
(10pts) Find a closed formula for the following sum:
z
3

z
5
+
z
7

z
9
+···
,
where

z
 <
1.
Solution.
This is an inﬁnite geometric series that can be written as
z
3
(1
+
q
+
q
2
+···
),
where
q
= 
z
2
. Therefore the result is
z
3
1

q
=
z
3
1
+
z
2
.
Problem 3
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 Fall '11
 FredPhelps
 Math

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