Question number 26. The first three elements of a certain sequence are 1, 2, and 3. Each element starting with the fourth is obtained by taking the sum of the two elements directly before the preceding element and subtracting from the sum the preceding element. Thus the sequence begins with 1, 2, 3 followed by 0, 5, minus 2, and 7, and so on. What is the 2010th element of this sequence? So let's write down the sequence and see if we can discover a pattern here for the general term. We have that A1 is 1, A2 is 2, A3 is 3. The fourth element, A4, is obtained by looking at the two elements directly before the preceding element, summing them, so that's 1 plus 2, and then subtracting the preceding element. That gives us 0. A5 is then the sum 2 plus 3 minus 0, so that is 5. A6 would be 3 plus 0 minus 5, which gives us minus 2. A7 would be 0 plus 5 minus the preceding element, which is negative 2, and that gives us 7. And we can keep going in this pattern. A8 would be obtained by summing 5 with negative 2 and then subtracting 7, so that gives us 3 minus 7, or negative 4. A9 is then negative 2 plus 7 minus negative 4, and that gives us exactly 5 minus negative 4, so 9. And now we start to notice a pattern. We notice that the odd numbers are always the same, so A sub odd is equal to the index odd. So, for example, A sub 2011 would be equal to 2011. A sub 2009 would be equal to 2009. And what we have to compute is A sub 2010. So what's the pattern for even numbers? And we can guess that pattern by looking at the even indices that we have. So A4 gives us 0, which is 4 less than 0. A6 gives us negative 2, which is the opposite of 2, and 2 is 4 less than 6. A8 gives us a negative 4 here. So it looks like the pattern is A here sub even would be equal to that even number minus 4, and then the answer is always negative. So we put a negative number here like that. And that certainly fits the pattern. That would tell us that A sub 10 is equal to negative 10 minus 4, or negative 6. And let's just check if that works. So A sub 10 would be the previous two elements before the preceding one, 7 plus negative 4, and then subtract 9, and that gives us 3 minus 9, which is negative 6. So the pattern seems to be OK. And if we use that pattern, then we have that A sub 2010 would be equal to the negative. We take 2010 and subtract 4 from it, and that gives us the answer as negative 2006. So that is choice A.

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