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Purple Comet Problems
2017 Purple Comet Problems
28
Purple Comet 2017 HS problem 28
Purple Comet 2017 HS problem 28
Source:
March 19, 2020
algebra
Sum
Problem Statement
Let
T
k
=
k
(
k
+
1
)
2
T_k = \frac{k(k+1)}{2}
T
k
=
2
k
(
k
+
1
)
be the
k
k
k
-th triangular number. The in finite series
∑
k
=
4
∞
1
(
T
k
−
1
−
1
)
(
T
k
−
1
)
(
T
k
+
1
−
1
)
\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}
k
=
4
∑
∞
(
T
k
−
1
−
1
)
(
T
k
−
1
)
(
T
k
+
1
−
1
)
1
has the value
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
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