Question

Identificand un factor comun in fiecare suma xalculati ;
1)3+6+9+...+66=
2)10+20+30+...+1230=
3)7+14+21+...+315=

Answer 1

$\displaystyle 1).3+6+9+...+66=3(1+2+3+...+22)=3 \cdot \frac{22(22+1)}{2} = \\ \\ =3 \cdot \frac{22 \cdot 23}{2} =3 \cdot \frac{506}{2} =3 \cdot 253=759 \\ \\ 2).10+20+30+...+1230=10(1+2+3+...+123)= \\ \\ =10 \cdot \frac{123(123+1)}{2} =10 \cdot \frac{123 \cdot 124}{2} =10 \cdot \frac{15252}{2} =10 \cdot 7626=76260 \\ \\ 3).7+14+21+...+315=7(1+2+3+...+45)=7 \cdot \frac{45(45+1)}{2} = \\ \\ =7 \cdot \frac{45 \cdot 46}{2} =7 \cdot \frac{2070}{2} =7 \cdot 1035=7245$