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Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009.
Induction
1. Prove that the sum of the ﬁrst
n
odd numbers is
n
2
.
2. Using the product rule
d
dx
(
f
(
x
)
g
(
x
)) =
f
0
(
x
)
g
(
x
) +
f
(
x
)
g
0
(
x
), and the fact that
d
dx
x
= 1, use induction to prove
that
d
dx
(
x
n
) =
nx
n

1
.
3. Let
a
n
be the sequence deﬁned by
a
1
=
√
2,
a
n
=
√
2
a
n

1
.
(a) Show that 1
< a
n
<
2 for all
n
.
(b) Show that
a
n
+1
> a
n
for all
n
.
NOTE: A sequence like this is called bounded and monotone and is guaranteed to converge.
4. The harmonic numbers are deﬁned by
H
n
= 1 +
1
2
+
1
3
+
1
4
+
···
+
1
n
.
(a) Show that
H
2
n
≥
1 +
n
2
for all
n
.
(b) Use your answer to (a) to show that
∞
X
n
=1
1
n
increases without bound.
5. Prove that 1 + 2
3
+ 3
3
+ 4
3
+
···
+
n
3
=
±
n
(
n
+ 1)
2
²
2
.
Strong Induction and Wellordering
1. Prove that if
n
≥
18, then you can make
n
cents out of just 4 and 7cent stamps
2. Here we’ll use wellordering to show that
x
2
+
y
2
= 3
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 Summer '09
 RobBayer
 Math, Product Rule

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