gelijk of niet gelijk, that is the question

Opgave - JWO 2006 dag 1 vraag 1

Bewijs dat $(1+m)(1+\frac{m}{2})(1+\frac{m}{2})\cdots(1+\frac{m}{n})=(1+n)(1+\frac{n}{2})(1+\frac{n}{3})\cdots(1+\frac{n}{m}) $

Oplossing

(met faculteiten)

$(1+m)(1+\frac{m}{2})(1+m/3)\cdots(1+m/n)=(1+n)(1+n/2)(1+n/3)\cdots(1+n/m)$
$\Leftrightarrow(1+m)(2/2+m/2)(3/3+m/3)...(n/n+m/n)=(1+n)(2/2+n/2)(3/3+n/3)...(m/m+n/m)$
$\Leftrightarrow(1+m)(2+m)(3+m)...(n+m)/(2*3*4*...*n)=(1+n)(2+n)(3+n)...(m+n)/(2*3*4*...*m)$
$\Leftrightarrow(1+m)(2+m)(3+m)...(n+m)/n!=(1+n)(2+n)(3+n)...(m+n)/m!$
$\Leftrightarrow m!(1+m)(2+m)(3+m)...(n+m)=n!(1+n)(2+n)(3+n)...(m+n)$
$<=>(m+n)!=(n+m)!$
$<=>m+n=n+m$