Presupunem prin absurd ca √2, √3 si √5 apartin aceeasi progresii aritmetice. Astfel, exisita m,n,p ∈ N, m≠n≠p astfel incat √2=a(m), √3=a(n) si √5=a(p)
√2=a1+(m-1)*r
√3=a1+(n-1)*r
√5=a1+(p-1)*r
√3-√2=(m-n)*r
√5-√2=(p-n)*r
==> [tex]\frac{\sqrt{3} -\sqrt{2} }{\sqrt{5} -\sqrt{2} } =\frac{m-n}{p-n}[/tex]
insa m-n/p-n apartine Q, dar cealalta nu, asa ca obtinem o contradictie..
deci cele 3 numere nu pot fi in aceeasi progresie aritmetica
