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 == | ||
Revision as of 21:40, 6 March 2015
Problem 2
Two geometric sequences 
and 
have the same common ratio, with 
, 
, and 
. Find 
.
Solution
Call the common 
Now since the 
th term of a geometric sequence with first term 
and common ratio 
is 
we see that 
But equals 
so 
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. 
