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).
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.
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Posted by Charlie
on 2016-10-08 09:32:36 |
D4? (spoiler)
