Question

Sa se calculeze suma 1+1/3+1/3^2+1/3^3+1/3^4

Answer 1

$1+\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\frac{1}{3^{4}}=$
$=\frac{3^{4}}{3^{4}}+\frac{3^{3}}{3^{4}}+\frac{3^{2}}{3^{4}}+\frac{3}{3^{4}}+\frac{1}{3^{4}}=$
$=\frac{1+3+3^{2}+3^{3}+3^{4}}{3^{4}}=$
$=\frac{\frac{3^{5}-1}{2}}{3^{4}}=$
$=\frac{3^{5}-1}{2*3^{4}}=$
$=\frac{243-1}{2*81}=$
$=\frac{242}{162}=\frac{121}{81}$