Question

se considera sumele:
S1=1+2+2²+2³+.....2(la puterea 2014)si
S2=1+3+3²+3³+....3(la puterea 2015).
a) verificati daca S1 se divide cu 7
b) verificati daca S2 se divide cu 40

determinati numerele abc cu bara deasupra astfel incat 15a+20c=16a

Answer 1

S₁=1+2+2²+2³+.....2²⁰¹⁴
S₁=(1+2+2²)+2³(1+2+2²)+...+2²⁰¹²(1+2+2²)
S₁=7+2³*7+...+2²⁰¹²*7
S₁=7(1+2³+...+2²⁰¹²) se divide cu 7

S₂=1+3+3²+3³+....3²⁰¹⁵
S₂=(1+3+3²+3³)+3⁴(1+3+3²+3³)+....+3²⁰¹²(1+3+3²+3³)
S₂=40+3⁴*40+...+3²⁰¹²*40
S₂=40(1+3⁴+...+3²⁰¹²)se divide cu 40