2002 AIME II Problems/Problem 11
Contents
Problem
Two distinct, real, infinite geometric series each have a sum of and have the same second term. The third term of one of the series is
, and the second term of both series can be written in the form
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
Solution 1
Let the second term of each series be . Then, the common ratio is
, and the first term is
.
So, the sum is . Thus,
.
The only solution in the appropriate form is . Therefore,
.
Solution 2
Let the two sequences be and
. We know for a fact that
. We also know that the sum of the first sequence =
, and the sum of the second sequence =
. Therefore we have
We can then replace
and
. We plug them into the two equations
and
. We then get
We subtract these equations, getting
Remember
, so
Then considering cases, we have either
or
. This suggests that the second sequence is in the form
, while the first sequence is in the form
Now we have that
and we also have that
. We can solve for
and the only appropriate value for
is
. All we want is the second term, which is
solution by jj_ca888
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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