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Let be a positive integer and define
MATHEMATICS · Subjective · AoPS · Mathematics · Functions · Functions
Problem
Let
n
n
n
be a positive integer and define
f
(
n
)
=
1
!
+
2
!
+
3
!
+
⋯
+
n
!
f(n) = 1! + 2! + 3! + \dots + n!
f
(
n
)
=
1
!
+
2
!
+
3
!
+
⋯
+
n
!
. Find polynomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
such that
f
(
n
+
2
)
=
Q
(
n
)
f
(
n
)
+
P
(
n
)
f
(
n
+
1
)
f(n+2) = Q(n)f(n) + P(n)f(n+1)
f
(
n
+
2
)
=
Q
(
n
)
f
(
n
)
+
P
(
n
)
f
(
n
+
1
)
for all
n
≥
1
n \ge 1
n
≥
1
.
Related theory
Sequences as Functions
Functions
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