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8
Sequence of Numbers
Sequence of Numbers
Source:
February 13, 2006
quadratics
algebra
polynomial
AMC
AIME
USA(J)MO
USAMO
Problem Statement
For any sequence of real numbers
A
=
(
a
1
,
a
2
,
a
3
,
…
)
A=(a_1,a_2,a_3,\ldots)
A
=
(
a
1
,
a
2
,
a
3
,
…
)
, define
Δ
A
\Delta A
Δ
A
to be the sequence
(
a
2
−
a
1
,
a
3
−
a
2
,
a
4
−
a
3
,
…
)
(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)
(
a
2
−
a
1
,
a
3
−
a
2
,
a
4
−
a
3
,
…
)
, whose
n
th
n^\text{th}
n
th
term is
a
n
+
1
−
a
n
a_{n+1}-a_n
a
n
+
1
−
a
n
. Suppose that all of the terms of the sequence
Δ
(
Δ
A
)
\Delta(\Delta A)
Δ
(
Δ
A
)
are
1
1
1
, and that
a
19
=
a
92
=
0
a_{19}=a_{92}=0
a
19
=
a
92
=
0
. Find
a
1
a_1
a
1
.
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