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jagr2713
 one year ago
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jagr2713
 one year ago
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jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0For a positive integer n, define as n degree polynomials\[f_{n}(x) \ as\] \[f_{n}(x)=\frac{ x(x1)(x2)...(xn+1) }{ n! }\] If the sum of all the roots of the functional equation \[f_{n}(x)+f _{n1}(x)=f _{n}(x+2)\] Is 8514, what is the value of n?

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0@ganeshie8 @nincompoop @geerky42

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Note that\[f_{n1} (x)=\frac{n}{xn+1} f_{n} (x) \] \[f_n(x+2)=\frac{(x+1)(x+2)}{(xn+1)(xn+2)} f_n (x)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0sum of all the roots of the functional equation?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0are you sure you don't mean sum of all the \(x\) satisfying that equation in \(x\) rather than talking about 'roots of a functional equation' (which would be presumably functions)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Put these back in the functional equation to get\[f_n(x) \left( 1+\frac{n}{xn+1} \frac{(x+1)(x+2)}{(xn+1)(xn+2)}\right)=0\]Simplify and find the roots, That's all I'm sayin, I'll say no more :)))

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0Hm i see :D So what is your answer

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0Igtg but i will see this later :D

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0$$\frac{(x)_n}{n!}+\frac{(x)_{n1}}{(n1)!}=\frac{(x+2)_n}{n!}\\(x)_n+n(x)_{n1}=(x+2)_n\\(x)_{n1}\left(xn+1+n\right)=(x+2)(x+1)(x)_{n2}\\(x)_{n2}(xn+2)(x+1)(x)_{n2}(x+1)(x+2)=0\\(x)_{n2}(x+1)\left[(xn+2)(x+2)\right]=0\\n(x)_{n2}(x+1)=0$$so let's find the sum of the roots of \((x)_{n2}=x(x1)(x2)\cdots(x(n2)+1)=x(x1)(x2)\cdots(x(n3))\):$$\sum_{k=0}^{n3}k=\frac12(n3)(n2)$$and then our additional root of \(1\) from \(x+1=x(1)\) gives us:$$\frac12(n2)(n3)1=8514\\(n2)(n3)=2\cdot8515 $$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0now notice \(\sqrt{2\cdot8515}\approx130.499\) so \(n2=131\implies n=133\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, the above should be correct; are these Brilliant problems or something

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0Correct @oldrin.bataku :D
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