[tex]\displaystyle a).6+11+16+21+...+371 \\ 371=6+(n-1) \times5 \\ 371=6+5n-5 \\ 5n=371-6+5 \\ 5n=370 \\ n=370:5 \\ n=74 \\ S_{74}= \frac{12+73 \times 5}{2} \times 74 \\ \\ S_{74}= \frac{12+365}{2} \times 74 \\ \\ S_{74}= \frac{377}{2} \times 74 \\ \\ S_{74}=377 \times 37 \\ S_{74}=13949[/tex]
[tex]\displaystyle b).5+12+19+26+33+..+705 \\ 705=5+(n-1) \times 7 \\ 705=5+7n-7 \\ 7n=705-5+7 \\ 7n=707 \\ n=707:7 \\ n=101 \\ S_{101}= \frac{10+100 \times 7}{2} \times 101 \\ \\ S_{101}= \frac{10+700}{2} \times 101 \\ \\ S_{101}= \frac{710}{2} \times 101 \\ \\ S_{101}=355 \times 101 \\ S_{101}=35855[/tex]
[tex]\displaystyle c).65+73+81+89+...+201 \\ 201=65+(n-1) \times 8 \\ 201=65+8n-8 \\ 8n=201-65+8 \\ 8n=144 \\ n=144:8 \\ n=18 \\ S_{18}= \frac{130+17 \times 8}{2} \times 18 \\ \\ S_{18}= \frac{130+136}{2} \times 18 \\ \\ S_{18}= \frac{266}{2} \times 18 \\ \\ S_{18}=133 \times 18 \\ S_{18}=2394[/tex]
[tex]\displaystyle d).123+128+133+138+...+1003 \\ 1003=123+(n-1) \times 5 \\ 1003=123+5n-5 \\ 5n=1003-123+5 \\ 5n=885 \\ n=885:5 \\ n=177 \\ S_{177}= \frac{246+176 \times 5}{2} \times 177 \\ \\ S_{177}= \frac{246+880}{2} \times 177 \\ \\ S_{177}= \frac{1126}{2} \times 177 \\ \\ S_{177}=563 \times 177 \\ S_{177}=99651[/tex]
