S=2×(1+2+3+....+2019)+2020
Aplicăm suma lui Gauss
2×(2019×2020:2)+2020
(2019×2020)+2020
2020×(2019+1)
2020×2020
scriem sub forma de putere
2020² => Este patrat perfect
Pătratul perfect presupune ridicarea unui număr la puterea a doua.
2 (1+2+3+...+2019)+2020-->aplic suma lui Gauss
2 [2019×(2019+1)÷2]+2020=4080400
2 (2019×2020÷2)+2020=4080400
2 (4078380÷2)+2020=4080400
2×2039190+2020=4080400
4078380+2020=4080400
4080400=2020 la puterea a doua, deci e pătrat perfect.
