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Tournament Of Towns
1990 Tournament Of Towns
(243) 1
TOT 243 1990 Spring O J1 identity with sum ( 1+ 1/2 + ...+ 1/n)^2
TOT 243 1990 Spring O J1 identity with sum ( 1+ 1/2 + ...+ 1/n)^2
Source:
March 12, 2021
algebra
Sum
Problem Statement
For every natural number
n
n
n
prove that
(
1
+
1
2
+
.
.
.
+
1
n
)
2
+
(
1
2
+
.
.
.
+
1
n
)
2
+
.
.
.
+
(
1
n
−
1
+
1
2
)
2
+
(
1
n
)
2
=
2
n
−
(
1
+
1
2
+
.
.
.
+
1
n
)
\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left( \frac{1}{n-1} + \frac12 \right)^2+ \left( \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)
(
1
+
2
1
+
...
+
n
1
)
2
+
(
2
1
+
...
+
n
1
)
2
+
...
+
(
n
−
1
1
+
2
1
)
2
+
(
n
1
)
2
=
2
n
−
(
1
+
2
1
+
...
+
n
1
)
(S. Manukian, Yerevan)
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