2011 AMC 12A Problems/Problem 8
Problem
In the eight term sequence , , , , , , , , the value of is and the sum of any three consecutive terms is . What is ?
Solution
Solution 1
Let . Then from , we find that . From , we then get that . Continuing this pattern, we find , , , and finally . So
Solution 2
A faster technique is to assume that the problem can be solved, and thus is an invariant. Since , assign any value to . is a simple value to plug in, which gives a value of for B. The 8-term sequence is thus . The sum of the first and the last terms is
Note that this alternate solution is not a proof. If the sum of had been asked for, this technique would have given as an answer, when the true answer would have been "cannot be determined".
Solution 3
Given that the sum of 3 consecutive terms is 30, we have and
It follows that because .
Subtracting, we have that .
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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All AMC 12 Problems and Solutions |
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