5 Arătaţi că x este număr rațional: ​

Răspuns:

Explicație pas cu pas:

1/1•2+1/2•3+...+1/24•25=

=1/1-1/2+1/2-1/3+....+1/24-1/25

=1/1-1/25

1/25•26+1/26•27+...+1/49•50=

=1/25-1/26+1/26-1/27+....+1/49-1/50

=1/25-1/50

1/94[1/1•2+1/2•3+...+1/24•25-(1/25•26+1/26•27+...+1/49•50)]=

=1/94[1/1-1/25-(1/25-1/50)]

=1/94(1/1-2/25+1/50)

=1/94[(50-4+1)/50]

=1/94·(47/50)

=1/2• 1/50

=1/100

x=√(1/100)=1/10 ∈Q

Salut!

1/[ n × (n + 1)] = 1/n - 1/(n + 1)

x = V1/94 × [(1/1 - 1/2 + 1/2 - 1/3 + ... + 1/24 - 1/25) - ( 1/25 - 1/26 + 1/26 - 1/27 + ... + 1/49 - 1/50 )]}

Se reduc cele cu semn opus.

x = V{1/94 × [(⁵⁰) 1/1 - ²) 1/25) - (²) 1/25 - 1/50)]}

x = V[1/94 × (48/50 - 1/50)]

x = V(1/94 × 47/50)

x = V(1/2 × 1/50) = V(1/100) = V1/V100 = 1/10 € Q