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National and Regional Contests
Canada Contests
Canada National Olympiad
1973 Canada National Olympiad
7
7
Part of
1973 Canada National Olympiad
Problems
(1)
Sum of fractions
Source: Canada 1973/7
1/11/2007
Observe that \frac{1}{1}= \frac{1}{2}+\frac{1}{2}; \frac{1}{2}=\frac{1}{3}+\frac{1}{6}; \frac{1}{3}=\frac{1}{4}+\frac{1}{12}; \frac{1}{4}= \frac{1}{5}+\frac{1}{20}. State a general law suggested by these examples, and prove it. Prove that for any integer
n
n
n
greater than 1 there exist positive integers
i
i
i
and
j
j
j
such that
1
n
=
1
i
(
i
+
1
)
+
1
(
i
+
1
)
(
i
+
2
)
+
1
(
i
+
2
)
(
i
+
3
)
+
⋯
+
1
j
(
j
+
1
)
.
\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}.
n
1
=
i
(
i
+
1
)
1
+
(
i
+
1
)
(
i
+
2
)
1
+
(
i
+
2
)
(
i
+
3
)
1
+
⋯
+
j
(
j
+
1
)
1
.
[hide="Remark."] It seems that this is a two-part problem.
induction