Difference between revisions of "2012 AIME II Problems/Problem 2"
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== Problem 2 == | == Problem 2 == | ||
− | Two geometric sequences <math>a_1, a_2, a_3, \ldots</math> and <math>b_1, b_2, b_3, \ldots</math> have the same common ratio, with <math>a_1 = 27</math>, <math>b_1=99</math>, and <math>a_{15}=b_{11}</math>. Find <math>a_9</math>. | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Two geometric sequences <math>a_1, a_2, a_3, \ldots</math> and <math>b_1, b_2, b_3, \ldots</math> have the same common ratio, with <math>a_1 = 27</math>, <math>b_1=99</math>, and <math>a_{15}=b_{11}</math>. Find <math>a_9</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
== Solution == | == Solution == | ||
− | Call the common <math>r.</math> Now since the <math>n</math>th term of a geometric sequence with first term <math>x</math> and common ratio <math>y</math> is <math>xy^{n-1},</math> we see that <math>a_1 \cdot r^{14} = b_1 \cdot r^{10} \ | + | Call the common ratio <math>r.</math> Now since the <math>n</math>th term of a geometric sequence with first term <math>x</math> and common ratio <math>y</math> is <math>xy^{n-1},</math> we see that <math>a_1 \cdot r^{14} = b_1 \cdot r^{10} \implies r^4 = \frac{99}{27} = \frac{11}{3}.</math> But <math>a_9</math> equals <math>a_1 \cdot r^8 = a_1 \cdot (r^4)^2=27\cdot {\left(\frac{11}{3}\right)}^2=27\cdot \frac{121} 9=\boxed{363}</math>. |
− | == See | + | ==Video Solution== |
+ | |||
+ | https://youtu.be/V2X9hz6DuUw | ||
+ | |||
+ | ~Lucas | ||
+ | |||
+ | == See Also == | ||
{{AIME box|year=2012|n=II|num-b=1|num-a=3}} | {{AIME box|year=2012|n=II|num-b=1|num-a=3}} | ||
+ | {{MAA Notice}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 17:32, 19 September 2022
Contents
Problem 2
Two geometric sequences and
have the same common ratio, with
,
, and
. Find
.
Solution
Call the common ratio Now since the
th term of a geometric sequence with first term
and common ratio
is
we see that
But
equals
.
Video Solution
~Lucas
See Also
2012 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.