Question

Arătaţi ca nr. 9 •(5+10+15+...+200):41 este pătrat perfect

Answer 1

5+10+15+...+200  Gauss

5(1+2+3+...+40)=
5*40*41/2=
5*20*41

reintroducem in ecuatie
9*5*20*41:41=
9*5*20=
900, e patrat perfect, pt ca radical din 900=30

Answer 2

$\displaystyle 9 \cdot (5+10+15+...+200):41=9 \cdot 5(1+2+3+...+40):41= \\ \\45(1+2+3+...+40):41=45 \cdot \frac{40(40+1)}{2} :41= \\ \\ =45 \cdot \frac{40 \cdot 41}{2} :41=45 \cdot \frac{1640}{2} :41=45 \cdot 820:41=45 \cdot 20=900=30^2$