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polynomial problem 3 (Posted on 2016-10-08) Difficulty: 4 of 5
Let P(x) be a nonzero polynomial such that

(x-1)P(x+1) = (x+2)P(x)

for every real x, and P(2)^2 = P(3).

Then find P(2016).

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution D4? (spoiler) | Comment 1 of 2
p(x+1) = (x+2)*p(x)/(x-1)

when x=2

p(3) = 4*p(2) = p(2)^2

p(2) = 4

p(3) = 16

Programming the transition from p(x) to p(x+1):


    5   open "poly3.txt" for output as #2
   10   X=3:P=16
   20   for Newx=4 to 2016
   30      P=P*(X+2)//(X-1)
   40      X=Newx:print X,P
   45      print #2,X,P
   50   next
   60   print X,P
   70   close #2

Adding third differences manually, shows a 3rd degree polynomial:

 x  P(x)       differences
 4   40 
 5   80      40
 6   140     60    20
 7   224     84    24    4
 8   336    112    28    4
 9   480    144    32    4
 10   660   180    36    4
 11   880 
 12   1144 
 13   1456 
 14   1820 
 15   2240 
 16   2720 
 17   3264 
 18   3876 
 19   4560 
 20   5320 
...

 2007   5389528224 
 2008   5397588336 
 2009   5405656480 
 2010   5413732660 
 2011   5421816880 
 2012   5429909144 
 2013   5438009456 
 2014   5446117820 
 2015   5454234240 
 2016   5462358720 
 
 The answer is  5462358720.
 


  Posted by Charlie on 2016-10-08 09:32:36
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