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Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1973 Canada National Olympiad
5
Harmonic
Harmonic
Source: Canada 1973/5
January 11, 2007
Problem Statement
For every positive integer
n
n
n
, let
h
(
n
)
=
1
+
1
2
+
1
3
+
⋯
+
1
n
.
h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.
h
(
n
)
=
1
+
2
1
+
3
1
+
⋯
+
n
1
.
For example,
h
(
1
)
=
1
h(1) = 1
h
(
1
)
=
1
,
h
(
2
)
=
1
+
1
2
h(2) = 1+\frac{1}{2}
h
(
2
)
=
1
+
2
1
,
h
(
3
)
=
1
+
1
2
+
1
3
h(3) = 1+\frac{1}{2}+\frac{1}{3}
h
(
3
)
=
1
+
2
1
+
3
1
. Prove that for
n
=
2
,
3
,
4
,
…
n=2,3,4,\ldots
n
=
2
,
3
,
4
,
…
n
+
h
(
1
)
+
h
(
2
)
+
h
(
3
)
+
⋯
+
h
(
n
−
1
)
=
n
h
(
n
)
.
n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).
n
+
h
(
1
)
+
h
(
2
)
+
h
(
3
)
+
⋯
+
h
(
n
−
1
)
=
nh
(
n
)
.
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