2002 AIME II Problems/Problem 11
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 .
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, .
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
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