Computing Value of a Functional Equation

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sakib17442
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Computing Value of a Functional Equation

Unread post by sakib17442 » Sun Jun 27, 2021 8:02 pm

A Function $f(x)$ is such that for any integer $x$, $f(x) + x f(2-x) = 6$ . Compute $-2019 f(2020)$
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Anindya Biswas
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Re: Computing Value of a Functional Equation

Unread post by Anindya Biswas » Tue Jun 29, 2021 3:25 am

sakib17442 wrote:
Sun Jun 27, 2021 8:02 pm
A Function $f(x)$ is such that for any integer $x$, $f(x) + x f(2-x) = 6$ . Compute $-2019 f(2020)$
$f(x)+xf(2-x)=6\cdots\cdots(1)$

Replacing $x\mapsto2-x$ gives,
$f(2-x)+(2-x)f(x)=6\cdots\cdots(2)$

Multiplying $x$ with $(2)$ and subtracting that from $(1)$ gives us,
$f(x)-(2x-x^2)f(x)=6-6x$
$\Rightarrow (1-x)^2f(x)=6(1-x)\cdots\cdots(3)$

Substituting $x=2020$ in $(3)$ gives us
$(-2019)^2f(2020)=6\cdot(-2019)$
$\Rightarrow\boxed{-2019f(2020)=6}$
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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