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2007 AIME Problems
12
Log of Geometric Sequence
Log of Geometric Sequence
Source: AIME II 2007 #12
March 29, 2007
calculus
integration
logarithms
ratio
AMC
USA(J)MO
USAMO
Problem Statement
The increasing geometric sequence
x
0
,
x
1
,
x
2
,
…
x_{0},x_{1},x_{2},\ldots
x
0
,
x
1
,
x
2
,
…
consists entirely of integral powers of
3.
3.
3.
Given that
∑
n
=
0
7
log
3
(
x
n
)
=
308
and
56
≤
log
3
(
∑
n
=
0
7
x
n
)
≤
57
,
\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,
n
=
0
∑
7
lo
g
3
(
x
n
)
=
308
and
56
≤
lo
g
3
(
n
=
0
∑
7
x
n
)
≤
57
,
find
log
3
(
x
14
)
.
\log_{3}(x_{14}).
lo
g
3
(
x
14
)
.
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