Question number 24. Julio creates a procedure for turning a set of three numbers into a new set of three numbers. He replaces each number by the sum of the other two. For example, the set containing the numbers 3, 4, and 6 becomes the set containing the numbers 10, 9, and 7. How many times must Julio apply this procedure to the set containing 1, 2, and 3 before he first obtains a set containing the number 2013? Let's study Julio's procedure by applying it several times. We begin with 1, 2, and 3. I'll write it as a sequence of three numbers. And then 1 must be replaced by 5, 2 by 4, and 3 is replaced by 3. Doing this again, now 5 is replaced by 7. We get 8 and 9. And applying the procedure again, 7 is replaced by 17. And let's do it one more time. 17 is replaced by the sum of 16 and 15. So each time we obtain three consecutive integers, and it would be nice if we could find a pattern. Let's look at the largest number in each sequence of three. And there is a pattern we can find. The largest numbers of each three-term sequence. Or of each iteration follow a pattern. We start with 3, and then if I multiply 3 and subtract 1, I do get 5, and then 5 times 2 minus 1 is indeed 9. 9 times 2 minus 1 is 17. And then 33 and so forth. So the pattern is the next term in the sequence is equal to twice the previous term and then minus 1. And so we can see that the terms increase pretty quickly. So let's just write down the next several terms until we get close to 2013 and see what happens. So after 33, I would have 65 and then 129, then 257, 513, 1025, and 2049, at which point I can stop because we see that we are not close to 2013. If 2013 were the smallest number in the set, then the largest number would have to be 2015, but we're not even close to that number. And so we can conclude that 2013 will never appear. So the answer is E.

number sequence

pattern

set transformation

extrapolation

impossible outcome