Question

Aratati ca numarul (5+10+15+....+200):41 este patrat perfect .

Answer 1

5+10+15+....+200=5(1+2+3+...+40)
1+2+3+...+40=(40x41)/2=820 - suma Gauss
5x820:41=
5x20=
100 sau 10², care este patrat perfect

Answer 2

$\displaystyle (5+10+15+....+200):41= \\ \\ \text{Factor comun in prima paranteza.} \\ \\ 5(1+2+3+...+40):41= \\ \\ \text{Se aplica Gaus in paranteza.} \\ \\ \boxed{\frac{n(n+1)}{2} }\ \texttt{n=ultimul nr din sir} \\ \\ \\ \frac{40(40+1)}{2}= \frac{\not40 \cdot 41}{\not2} = 20 \cdot 41 = 820 \\ \\ 5 \cdot 820:41= \\ \\ 4100:41= \\ \\ =100 \\ \\ \boxed{\boxed{\bold{100=10^2}}}$