Question

dovediti ca:a) 1+2+2 la puterea a doua + 2 la puterea a treia +...+63= 2 la puterea 64 -1

Answer 1

^=la puterea
S=2^0+2^1+2^2+2^3+.....+2^63
Inmultind cu 2 obtinem:
2·S=2^0*2+2^1*2+2^2*2+2^3*2+.....+2^63*2
2·S=2^(0+1)+2^(1+1)+2^(2+1)+2^(3+1)+.....+2^(63+1)
2·S=2^1+2^2+2^3+2^4+......+2^64
2·S=(2^0+2^1+2^2+2^3+....+2^63)+2^64-2^0
Asadar:
2·S=S+2^64-2^0
De unde:
S=2^64-1