Question

Aratati ca numarul a=(x*x²*x³*...*x la puterea 20) : (x*x³*x la puterea 5*...* x la puterea 19 este p.p. (*=ori)

Answer 1

a=$(x * x^{2} * x^{3} *...* x^{20} ):(x* x^{3} * x^{5} *...* x^{19} )$=

=$x^{1+2+3+...+20} : x^{1+3+5+...+19}$ 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=$x^{20*21:2} : x^{10*10}$=

=$x^{10*21} : x^{10*10}$=

=$x^{10*21-10*10}$=

=$x^{10*(21-10)}$=

=$x^{10*11}$=

=$x^{11*5*2}$=

=$x^{55*2}$=

$( x^{55} )^{2}$ care este patrat perfect.