Question

Fie S=1+3+3^2+3^3+...+3^80.
 a)Aratati ca S este divizibil cu 13.
 b)Demonstrati ca 2*S>27^27-9.

Answer 1

S= (1+3+3²)+3³(1+3+3²) +............+3^78(1+3+3²) = 13+13·3³+..........13·3^78 = 13(1+3³+.........+3^78) ⇔divizibil cu13

2S>27^27 - 9 ⇔ 2S > 3^81 - 3²                                                                   3S = 3+3²+3³+......3^81          3S - S = 2S = 3^81 - 1       si, deoarece    (3^81 - 1) > (3^81 - 9) ⇒ 2S >3^81 - 9