Application of Equations #2

Seems like you'll need a lot of calculations, but actually you calculate only $21$ times.

Let's define $f(x)=ax+bx^2+cx^3+dx^4+ex^5$ .

Then $f(0)=0$ , $f(1)=1$ , $f(2)=3$ , $f(3)=4$ , $f(4)=9$ , $f(5)=x$ , $f(6)=226$ .

Generally, performing the calculation $f(n+1)-f(n)$ decreases the degree of the function by $1$ .

Since $f(x)$ is a function of 5th degree, it becomes an invariable when we perform the calculation $5$ times.

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First, I'll let you know how I expressed the calculation.

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$\begin{array} {ccccc} \boxed{a}&&\boxed{b}&&\boxed{c} \\ &\boxed{b-a}&&\boxed{c-b}& \\ &&\boxed{c-2b+a}&& \end{array}$

Good, now let's begin.

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$\begin{array} {ccccccccccccc} \boxed{0}&&\boxed{1}&&\boxed{3}&&\boxed{4}&&\boxed{9}&&\boxed{x}&&\boxed{254} \\ &\boxed{1}&&\boxed{2}&&\boxed{1}&&\boxed{5}&&\boxed{x-9}&&\boxed{254-x}& \\ &&\boxed{1}&&\boxed{-1}&&\boxed{4}&&\boxed{x-14}&&\boxed{263-2x}&& \\ &&&\boxed{-2}&&\boxed{5}&&\boxed{x-18}&&\boxed{277-3x}&&& \\ &&&&\boxed{7}&&\boxed{x-23}&&\boxed{295-4x}&&&& \\ &&&&&\boxed{x-30}&&\boxed{318-5x}&&&&& \end{array}$

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$x-30=318-5x$ ,

$\boxed{\therefore x=58}$