Selected Solutions to Homework # 4
1. # 4 in Appendix C: Let
r
be a real number,
r
6
= 1. Prove that for
every
n
∈
Z
+
,
1 +
r
+
r
2
+
···
+
r
n

1
=
r
n

1
r

1
.
Proof: We give a proof using induction. Let
r
∈
R
and
r
6
= 1. Let
P
n
be the statement that the above equation holds for the integer
n
. The
statement
P
1
reads 1 =
r

1
r

1
, which is evidently true. Suppose that
the statement
P
n
is true for some
n
≥
1. We are to show that the
statement
P
n
+1
is true. Consider 1 + 2 +
···
+
r
n

1
+
r
n
. By grouping
the ﬁrst
n
terms and applying
P
n
, we have
1+2+
···
+
r
n

1
+
r
n
=
r
n

1
r

1
+
r
n
=
r
n

1
r

1
+
r
n
(
r

1)
r

1
=
r
n
+1

1
r

1
.
Thus we have shown that
P
n
+1
is a consequence of
P
n
. By the principle
of mathematical induction,
P
n
is true for every integer
n
≥
1. Q.E.D.
2. # 16 in Appendix C. This is the game known as the “Towers of Hanoi”.
I will give a brief sketch of how to solve this problem.
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 Spring '10
 R.Bell
 Algebra, Natural number, Greatest common divisor, Euclidean algorithm, PEG

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