a=[tex](x * x^{2} * x^{3} *...* x^{20} ):(x* x^{3} * x^{5} *...* x^{19} )[/tex]=
=[tex] x^{1+2+3+...+20} : x^{1+3+5+...+19} [/tex] Folosim suma Gauss si formula sumei numerelor impare consecutive:
SG=1+2+3+...+n=n*(n+1):2
SI=1+3+5+...+(2k-1)=k*k
a=[tex] x^{20*21:2} : x^{10*10} [/tex]=
=[tex] x^{10*21} : x^{10*10} [/tex]=
=[tex] x^{10*21-10*10} [/tex]=
=[tex] x^{10*(21-10)} [/tex]=
=[tex] x^{10*11} [/tex]=
=[tex] x^{11*5*2} [/tex]=
=[tex] x^{55*2} [/tex]=
[tex] ( x^{55} )^{2} [/tex] care este patrat perfect.
