Question

se considera sumele S1=1+2+2^2+...+2^2011 si S2=1+2+3+...+2009. Aratati ca diferenta S1-S2 este divizibila cu 10

Answer 1

      
U = ultima cifra

$S1-S2= (2^{0}+2^{1}+2^{2}+.....+2^{2011})-(1+2+3+.....+2009)= \\ \\ =(2^{2012}-1)- (\frac{2009*2010}{2})= \\ \\ (2^{2012}-1)- (2009*1005) \\ \\------ \\ \\ U((2^{2012}-1)- (2009*1005)) = \\ U(2^{2012}-1)-U(2009*1005)= \\ U(2^{4*503}-1)-5=U[( 2^{4})^{503}]-1-5=U(16^{503})-6 = 6-6=0 \\ \\ => (S1 - S2) \,\,\,este\,\,\,divizibila\,\,\,cu\,\,\,10$