Question

Calculeaza : a] 3+7+11+15+.........+119=
b] 6+11+16+21+.......+371=
c] 5+12+19+26+33+.......+705
Va rog

Answer 1

$\displaystyle a).3+7+11+15+...+119 \\ \\ 119=3+(n-1) \times 4 \\ \\ 119=3+4n-4 \\ \\ 4n=119-3+4 \\ \\ 4n=120 \\ \\ n= \frac{120}{4} \\ \\ n=30$

$\displaystyle S_3_0= \frac{6+29 \times 4}{2} \times 30 \\ \\ S_3_0= \frac{6+116}{2} \times 30 \\ \\ S_3_0= \frac{122}{2} \times 30 \\ \\ S_{30}=61 \times 30 \\ \\ S_3_0=1830$

$\displaystyle b). 6+11+16+21+...+371 \\ \\ 371=6+(n-1) \times 5 \\ \\ 371=6+5n-5 \\ \\ 5n=371-6+5 \\ \\ 5n=370 \\ \\ n= \frac{370}{5} \\ \\ n=74$

$\displaystyle S_{74}= \frac{12+73 \times 5}{2} \times 74 \\ \\ S_{74}= \frac{12+365}{2} \times 74 \\ \\ S_{74}= \frac{377}{2} \times 74 \\ \\ S_{74}=377 \times 37 \\ \\ S_{74}=13949$

$\displaystyle c).5+12+19+26+33+...+705 \\ \\ 705=5+(n-1) \times 7 \\ \\ 705=5+7n-7 \\ \\ 7n=705-5+7 \\ \\ 7n=707 \\ \\ n= \frac{707}{7} \\ \\ n=101$

$\displaystyle S_{101}= \frac{10+100 \times 7}{2} \times 101 \\ \\ S_{101}= \frac{10+700}{2} \times 101 \\ \\ S_{101}= \frac{710}{2} \times 101 \\ \\ S_{101}=355 \times 101 \\ \\ S_{101}=35855$