Question

Calculati

2×3+2×3 la puterea 2+...+2×3 la puterea 20
2+2 la puterea 2+2 la puterea 3+2 la puterea 4+2 la puterea 5+2 la puterea 6+2 la puterea 7 +2 la puterea 8
2+2 la puterea 2+2 la puterea 3 +2 la puterea 4+...+2 la puterea 20

2 +2 la puterea 2 +2 la puterea 3+2 la puterea 4+...+ 2 la puterea 200
Va rog...

Answer 1

la primul exercitiu nu am inteles ce se ridica la putere
$[tex]2+ 2^{2} +2^{3}+2^{4}+...........}+2^{20}=$[/tex]
adaug un doi in fata si scad un doi la sfarsit:
$2+2+ 2^{2} +2^{3}+2^{4}+2^{5}+2^{6}+2^{7}+2^{8}-2=$
primii 2 termeni ii adun:
$2^{2}+ 2^{2} +2^{3}+2^{4}+2^{5}+2^{6}+2^{7}+2^{8}-2=$
$2*2^{2} +2^{3}+2^{4}+2^{5}+2^{6}+2^{7}+2^{8}-2=$
$2^{3} +2^{3}+2^{4}+2^{5}+2^{6}+2^{7}+2^{8}-2=$
$2^{4}+2^{4}+2^{5}+2^{6}+2^{7}+2^{8}-2=$
.................................
$2^{8}+2^{8}-2=2^{9}-2$
dupa aceeasi regula adaug un doi in fata si scad un doi la sfarsit:
$2+2+ 2^{2} +2^{3}+2^{4}+..........+2^{20}-2=$
$2^{2} + 2^{2} +2^{3}+2^{4}+..........+2^{20}-2=$
$2^{3} +2^{3}+2^{4}+..........+2^{20}-2=$
.........
$2^{20} +2^{20}-2=2^{21}-2$
dupa acceasi regula se face ultimul exercitiu 
rezultatul este 
$2^{201}-2$