Question

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

Answer 1

9*5*(1+2+3+...+40):41=9*5*$\frac{40\cdot\not{41^1}}{2}$:$\not{41}_{1}$=\\
=9*5*4*5*=(2*3*5)² =30²

Answer 2

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