Question

Aratati ca numarul a=(x·$x^{2}$·x³·...·$x^{20}$)÷(x·x³·$x^{5}$·...·$x^{19}$) este patrat perfect, oricat ar fi numarul natural nenul x.

Answer 1

x . x^2 . x^3. ..........x^20=x^(1+2+3+.............+20)
1+2+3+..................+20=(1+20) x 20/2=210
x . x^3 . x^5 .................x^19 =x^(1+3+5+.........+19)
1+3+5+.............19=(1+19)x10/2 = 100
(x^210)/x^100=x^110=(x^55)^2 deci e patrat perfect