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2021 MMATHS
5
MMATHS 2021, Problem 5: A Not-Quite Fib
MMATHS 2021, Problem 5: A Not-Quite Fib
Source:
October 31, 2021
Yale
MMATHS
Problem Statement
Suppose that
a
1
=
1
a_1 = 1
a
1
=
1
, and that for all
n
≥
2
n \ge 2
n
≥
2
,
a
n
=
a
n
−
1
+
2
a
n
−
2
+
3
a
n
−
3
+
…
+
(
n
−
1
)
a
1
.
a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.
a
n
=
a
n
−
1
+
2
a
n
−
2
+
3
a
n
−
3
+
…
+
(
n
−
1
)
a
1
.
Suppose furthermore that
b
n
=
a
1
+
a
2
+
…
+
a
n
b_n = a_1 + a_2 + \ldots + a_n
b
n
=
a
1
+
a
2
+
…
+
a
n
for all
n
n
n
. If
b
1
+
b
2
+
b
3
+
…
+
b
2021
=
a
k
b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k
b
1
+
b
2
+
b
3
+
…
+
b
2021
=
a
k
for some
k
k
k
, find
k
k
k
.Proposed by Andrew Wu
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