Since the previous one was solved so quickly, I figured I'd put up a new one. Here are the first ten terms of a sequence. 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, . . . Same idea as yesterday, describe the pattern of the sequence. Also, since I'm not stating it this time, what would the magical curve of best fit for this one do at infinity? What would its limit be? What would its derivative's limit be? If you use a search engine, please don't submit it in the comments. Thursday night, I'll post in the comments section the solution. If you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.
Tuesday, February 24, 2009
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4 comments:
Haha, I enjoyed the analogy in the title.
I believe the next 5 terms in the sequence are
83, 98, 114, 131, 150,...
where 83 = 69 + 14, 98 = 83 + 15, 114 = 98 + 16, and so on, so that the second term of each addition increases by 1 along the sequence - unless that second term has already appeared in the sequence, in which case the term is increased by 2 instead (as in the case going from 131 to 150).
The above sequence (let's call it sequence A) is similar to the follow sequence (sequence B):
1, 3, 6, 10, 15, 21,...
which would have the "magical curve" of 0.5n^2 + 0.5n for the n-th term, whose limit at infinity is infinity. Its derivative's limit, being a linear curve, is also infinity at infinity. Since sequence A is increasing at a slightly faster rate than B (compare the terms after the 2nd term), the limit of sequence A and the limit of the derivative of its magical curve would also both be infinity.
Right?
I think you're funny, KWTsui, for thanking me for the suggestion and then not using it.
That's a good formal proof that the derivative goes to infinity.
And to comment on your use of "magical curve" I've determined that "mystical" probably would have been more appropriate than "magical." For your sequence, where it's the n-1 term plus n, you doesn't have a magical curve; it has a real curve of best fit. Just like how the Fibonacci doesn't have a mystical curve of best fit, it has a real, existing one.
I also find it humourous that you ask "Right?" at the end. You know you're right!
You should comment more often on the blog since you add so much amusement to it. Thank you :)
Hey, I did you your suggestion... look, no link to my blog! And since I'm already on as kwtsui, I might as well use it.
*I did use your suggestion*
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