Question

Calculati: (1- 1/3 + 1/3^2 - 1/3^3 +...- 1/3^2011) : (1- 1/3^2012)

Answer 1

$(1+ \frac{1}{3} - \frac{1}{ 3^{2} } +...- \frac{1}{ 3^{2011} } ): \frac{1}{3^{2012} } =$
$(1+ \frac{1}{3} - \frac{1}{ 3^{2} } +...- \frac{1}{ 3^{2011} } )* 3^{2012} } =$
$3^{2012} -3^{2011}*2+3^{2009}*2+...+3*2=$
$2*(3^{2011}+3^{2009}+...+3^{1})=$
Notez cu S:
$S=3^{2011}+3^{2009}+...+3^{1}$
Inmultesc cu $3^{2}$
$3^{2} *S=3^{2013}+3^{2011}+...+3^{3}+3^{1}-3^{1}$
$3^{2} S=3^{2013}+S-3^{1}$
$S=(3^{2013}-3)/(3^{2} -1)$
NUMARUL CAUTAT:=2*S=$(3^{2013} -3)/4$