Question

Calculati 1+2+3+.................100 1+2+3+.............1001 1+2+3+................ 560 3+6+9+............. 303 5+10+15+.........................+305 care raspunde ? 

Answer 1

$\displaystyle 1+2+3+...+100= \frac{100(100+1)}{2} = \frac{100 \times 101}{2} = \frac{10100}{2} =5050 \\ \\ 1+2+3+...+1001= \frac{1001(1001+1)}{2} = \frac{1001 \times 1002}{2} = \\ =\frac{1003002}{2} =501501 \\ \\ 1+2+3+...+560= \frac{560(560+1)}{2} = \frac{560 \times 561}{2} = \frac{314160}{2} =157080$

$\displaystyle 3+6+9+...+303=3(1+2+3+...+101)=3 \times \frac{101(101+1)}{2} = \\ \\ =3 \times \frac{101 \times 102}{2} =3 \times \frac{10302}{2} =3 \times 5151=15453 \\ \\ 5+10+15+...+305=5(1+2+3+...+61)= 5 \times \frac{61(61+1)}{2} = \\ \\ =5 \times \frac{61 \times 62}{2} =5 \times \frac{3782}{2} =5 \times 1891=9455$