1. The first time I listed the first few terms: \|2 +1, 3 + 2\|2, 7 + 5\|2, 17 + 12\|2, ... because \|2 is irrational, I have no way to figure out the pattern. I was stuck!
2. 500/501 is too relaxed because (\|2 + 1)^3 = 14.067334..., so 300/301 will have the same answer.
3. The most important point. What is behind these two puzzles? What is the point? The point is that (\|2 +1)^n is infinitely approaching closer to some intergers from both sides alternatively ( > < > <...) after n is big enough. How can I find this pattern? If I really caculated this sequense, take \|2 = 2.4, i will have 2.4, 5.76, 13.824, 33.1776, 79.626, 191.103, 458.647, 1100.753, 2641.808, 6340.3381, 15216.811, 36520.347, 87648.834, 210357.2, 504857.28, 1211657.5 .... I still can not see that pattern. So the point is we have to find the trick to know the twin numbers relation. That is hard for me because I am not keen to this special number if I haven't worked on many many similar questions.
When I was in middle school, I was often overwhelmed by the tricks I read from books. If I cannot find the real meaning of the math problem that can really hurt my interest and confidence. So I always told my son "any smartest answer came from the dumbest and very normal way of thinking". I wish a kid's math thinking be a stream of fluent not blocked by fear and lack of confidence. Last year my son had a math problem " s = 8 +88 +888 + 8888 + ... + 88888888, find the last four digits of s". He said "this is hard. I don't know how to do it". I told him "just do the normal addition". He really added one by one, but just after the first three, he instantly figure out the quickest way to do it. He added eight 8s, saved the tenth digit and added 7 8's with the saved tenth digit from last time, and so on. I told him again "every smart way came from the dumbest way".
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