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How can I calculate this: ($\dots$(((2 * 3) * 4) * 5) * $\dots$) * 2017 ; if x * y = $\frac{x + y}{1 + xy}$, (notice x*y is not multiplication). This should be calculated using recursive formulas only.

nivalen292
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2 Answers2

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Note that $ \star $ is associative \begin{eqnarray*} x \star y \star z = \frac{x+y+z+xyz}{1+xy+yz+zx}. \end{eqnarray*} Now if \begin{eqnarray*} g(X)=(X-x)(X-y) (X-z) =X^3+X(xy+yz+zx)-(X^2(x+y+z)+xyz) \end{eqnarray*} then \begin{eqnarray*} g(1) =1+(xy+yz+zx)-((x+y+z)+xyz) \\ g(-1)=1+(xy+yz+zx)+((x+y+z)+xyz) \\ \end{eqnarray*} so \begin{eqnarray*} x \star y \star z = \frac{x+y+z+xyz}{1+xy+yz+zx} =\frac{g(-1)-g(1)}{g(-1)+g(1)} \end{eqnarray*} And a similar arguement will work for more iterands.

Now define \begin{eqnarray*} f(X)=(X-2)(X-3) \cdots (X-2017) \end{eqnarray*} then \begin{eqnarray*} 2 \star 3 \cdots 2016 \star 2017 = \frac{f(-1) -f(1)}{f(-1)+f(1)} =\frac{2018! -2 \times 2016!}{2018! + 2 \times 2016!}. \end{eqnarray*}

Donald Splutterwit
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  • Could you please give a more detailed explanation on your solution. What does X equal to and why are -1 and 1 the arguments to the function. – nivalen292 Dec 31 '17 at 18:44
  • [+1] Besides, it can be connected to $\tan$ and $\arctan$ functions. See for example (https://math.stackexchange.com/q/1876639). – Jean Marie Mar 07 '18 at 01:11
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Note: $$x*y*z=x*z*y$$ $$2*4=\frac23; 2*4*6*8=\frac45; \cdots 2*4*\cdots *2014*2016=\frac{1008}{1009}.$$ $$3*5=\frac12; 3*5*7*9=\frac23; \cdots 3*5*\cdots *2015*2017=\frac{504}{505} $$ $$\frac{1008}{1009}*\frac{504}{505}=...$$

farruhota
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