[tex]\displaystyle a).(12+24+36+...+444):(6 +12+18+...+222)= \\ \\ =12(1+2+3+...+37):6(1+2+3+...+37)=12:6=2[/tex]
[tex]\displaystyle b).(4+8+12+....+2012):(2+4+6+...+1006)= \\ \\ =4(1+2+3+...+503):2(1+2+3+...+503)=4:2=2[/tex]
[tex]\displaystyle c).6+11+16+21+...+506 \\ 506=6+(n-1) \times 5 \\ 506=6+5n-5 \\ 5n=506-6+5 \\ 5n=505 \\ n=505:5 \\ n=101 \\ S_{101}= \frac{12+100 \times 5}{2} \times 101 \\ \\ S_{101}= \frac{12+500}{2} \times 101 \\ \\ S_{101}= \frac{512}{2} \times 101 \\ \\ S_{101}=256 \times 101 \\ S_{101}=25856[/tex]
[tex]\displaystyle d).4+9+14+19+...+1004 \\ 1004=4+(n-1) \times 5 \\ 1004=4+5n-5 \\ 5n=1004-4+5 \\ 5n=1005 \\ n=1005:5 \\ n=201 \\ S_{201}= \frac{8+200 \times 5}{2} \times 201 \\ \\ S_{201}= \frac{8+1000}{2} \times 201 \\ \\ S_{201}= \frac{1008}{2} \times 201 \\ \\ S_{201}=504 \times 201 \\ S_{201}=101304[/tex]
