Răspuns:
[√ (n²+1)]+ [√ (n²+2)]+ [√ (n²+3)]+…..+ [√ (n²+n)]=n², este pătrat perfect
Explicație pas cu pas:
[√ (n²+1)]+ [√ (n²+2)]+ [√ (n²+3)]+…..+ [√ (n²+n)], n∈N*
n²+n=n(n+1)
[√ (n*n)]< [√ (n(n+1)]<[√ (n+1)²]
n <[√ (n(n+1)] < n+1
[√ (n²+1)]=[√ (n²+2)]= [√ (n²+3)]=…..= [√ (n²+n)]=n
=>[√ (n²+1)]+ [√ (n²+2)]+ [√ (n²+3)]+…..+ [√ (n²+n)]=
=n+n+n+…+n=n*n=n² este pătrat perfect
