[tex]\displaystyle a).4+8+12+...+2000=4(1+2+3+...+500)= \\ \\ =4 \times \frac{500(500+1)}{2} =4 \times \frac{500 \times 501}{2} =4 \times \frac{250500}{2} = \\ \\ =4 \times 125250=501000 [/tex]
[tex]\displaystyle b).5+10+15+...+2000=5(1+2+3+...+400)= \\ \\ =5 \times \frac{400(400+1)}{2} =5 \times \frac{400 \times 401}{2} =5 \times \frac{160400}{2} = \\ \\ =5 \times 80200=401000[/tex]
[tex]\displaystyle c).7+14+21+...+2002=7(1+2+3+...+286)= \\ \\ =7 \times \frac{286(286+1)}{2} =7 \times \frac{286 \times 287}{2} =7 \times \frac{82082}{2} = \\ \\ =7 \times 41041=287287[/tex]
[tex]\displaystyle d).17+34+51+...+2006=17(1+2+3+...+118)= \\ \\ = 17 \times \frac{118(118+1)}{2} =17 \times \frac{118 \times 119}{2} =17 \times \frac{14042}{2} = \\ \\ =17 \times 7021=119357[/tex]
