Difference between revisions of "2007 AIME II Problems/Problem 12"
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<math>\sum_{n=0}^{7}\log_{3}(x_{n}) = 308</math> and <math>56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,</math> | <math>\sum_{n=0}^{7}\log_{3}(x_{n}) = 308</math> and <math>56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,</math> | ||
− | find <math> | + | find <math>\log_{3}(x_{14}).</math> |
== Solution == | == Solution == | ||
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[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 23:33, 4 July 2013
Problem
The increasing geometric sequence consists entirely of integral powers of
Given that
and
find
Solution
Suppose that , and that the common ratio between the terms is
.
The first conditions tells us that . Using the rules of logarithms, we can simplify that to
. Thus,
. Since all of the terms of the geometric sequence are integral powers of
, we know that both
and
must be powers of 3. Denote
and
. We find that
. The possible positive integral pairs of
are
.
The second condition tells us that . Using the sum formula for a geometric series and substituting
and
, this simplifies to
. The fractional part
. Thus, we need
. Checking the pairs above, only
is close.
Our solution is therefore .
See also
2007 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.