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Maths Pattern Problem 
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#1
Aug1414, 02:50 AM

P: 117

3,8,17,24,49,58,117,?
what is the missing number ? Also give the pattern you followed. 


#2
Aug1414, 05:59 AM

P: 3,099

Is this a homework assignment?



#3
Aug1414, 06:36 AM

P: 117

Nope it is not ,i came across this question in a book and i am curious about its solution but i am not able to find a pattern series is following.



#4
Aug1414, 07:24 AM

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Maths Pattern Problem
There are of course, an infinite number of solutions. One way, pretty much "brute strength" is to find a polynomial, P(n), such that P(0)= 3, P(1)= 8, P(2)= 17, P(3)= 24, P(4)= 49,P(5)= 58, P(6)=117, and then calculate P(7). Since we are given 8 values, there is a unique 7th degree polynomial satisfying that.
We can find that polynomial using the "difference". If we subtract each number in the sequence from the next we get the 7 differences 8 3= 5, 17 8= 9, 24 17= 7, 49 24= 25, 58 49= 9, and 117 58= 59. The second differences are 9 5= 4, 7 9= 2, 25 7= 18, 925= 16, and 59 9= 50. The third differences are 2 4= 6, 18 (2)= 20, 16 18= 34, and 50 (16)= 66. The fourth differences are 20 (6)= 26, 34 20= 54, 66 (34)= 100. The fifth differences are 54 26= 80 and 100 (54)= 154. Finally, the sixth difference is 154 (80)= 234. Now, by "Newton's difference formula, the polynomial that, taking n to be 0 to 6, gives those values is 8+ 9n+ (4/2!)n(n 1) (6/3!)n(n1)(n2)+ (26/4!)n(n 1)(n 2)(n 3) (30/5!)n(n 1)(n 2)(n 3)(n 4)+ (184/6!)n(n 1)(n 2)(n 3)(n 4)(n 5). Set n= 7 in that polynomial to determine the next number in that sequence. 


#5
Aug1414, 07:32 AM

P: 3,099

Wow Halls impressive! I never knew about Newton's difference formula. I tried the differences to the 3rd order but still didn't see a pattern and began to think the sequence was wrong or from the Veritasium video something completely unexpected:
https://www.youtube.com/watch?v=vKA4w2O61Xo 


#6
Aug1414, 08:25 AM

P: 545

128.
3 + 5 = 8 8 x 2 + 1 = 17 17 + 7 = 24 24 x 2 + 1 = 49 49 + 9 = 58 58 x 2 + 1 = 117 117 + 11 = 128 


#7
Aug1414, 08:31 AM

P: 545

As you well know HallsofIvy 


#8
Aug1414, 09:54 AM

P: 3,099

Using Occam's razor is still an arbitrary choice for any problem with incomplete knowledge. There may yet be an underlying pattern to the one you discovered that even more understandable using one rule instead of two. The veratasium video highlights that premise where the rule was totally unexpected. http://en.wikipedia.org/wiki/Occam_razor Also could we drop the sarcasm from your post? Halls is a respected contributor and mentor to this forum and part of his responsibility is to direct students along the path of solution but not actually solve it. 


#9
Aug1414, 09:57 AM

P: 3,099

How did you come by your solution? What insight did you have or what method did you follow? 


#10
Aug1414, 11:11 AM

P: 545




#11
Aug1414, 11:25 AM

P: 999

The pairs occur in a pattern (every odd numbered term is the first member of such a pair). The first members of those pairs occur in a pattern (simple arithmetic sequence of differences). 


#12
Aug1414, 11:39 AM

P: 3,099

My apologies, I didn't understand that your sarcasm was in response to Hall's solution which was perceived as sarcasm. Personally, I felt it was complex but it brought to light another way to solve these kinds of problems.
As I looked at it I did see the odd numbers 5, 7, and 9 so perhaps that's all that needed to solve it. However, I didn't see the succ = pred*2 + 1 expression though. Can you tell us how you came to your solution? It would help the OP understand the methods of solution better. I didn't see the complete solution either so I too would benefit. Thanks. 


#13
Aug1414, 01:24 PM

P: 117

Thanks a lot :)



#14
Aug1414, 05:07 PM

P: 545

In terms of useful general methods of solution, looking at differences between terms is always the place to start. When you are looking at this:
I would stress that, like jbriggs444, I don't think this is the complete solution: if it is, it is rather unsatisfactory. 


#15
Aug1414, 07:20 PM

P: 3,099

So you did the same analysis that we did and had a flash of insight.
WHats interesting is that while we saw the odd number progression we ignored it while we searched for a more inclusive algorithm. Its a classic 'cant see the forest for the trees' problem we only needed the odd number progression to answer the problem. It reminds me of a famous code breaking movie where the protagonist was trying to crack a Russian coded transmission. He used every sort of codebreakers trick but came up empty handed. The signal sounded like beads in a bottle sloshing around. His insight came when he gave up for the night and sat down with his infant son and shook the baby rattle. The code was the shakng of the rattle not the more intricate sounds of the beads shaking about. And the movie is: http://en.wikipedia.org/wiki/Sebastian_%281968_film%29 now showing on youtube at your earliest convenience: https://www.youtube.com/watch?v=bIK3OYnD9MY 


#16
Aug1414, 11:53 PM

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