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What is the following value when expressed as a common fraction: $$\frac{1}{3^{1}}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}?$$

Guest Jan 25, 2015

Best Answer 

 #1
avatar+92805 
+10

GP

a=1/3     r= 1/3     n=6

 

$$\\\boxed{S_n=\frac{a(1-r^n)}{1-r}}\\\\
S_6=\frac{\frac{1}{3}(1-(\frac{1}{3})^6)}{1-\frac{1}{3}}}\\\\
S_6=\frac{\frac{1}{3}(1-\frac{1}{3^6})}{\frac{2}{3}}}\\\\
S_6=\frac{1}{\not{3}}(1-\frac{1}{3^6})\times \frac{\not{3}}{2}\\\\
S_6=\frac{1}{2}(1-\frac{1}{3^6})\\\\$$

 

$${\mathtt{0.5}}{\mathtt{\,\times\,}}\left({\mathtt{1}}{\mathtt{\,-\,}}\left({\frac{{\mathtt{1}}}{{{\mathtt{3}}}^{{\mathtt{6}}}}}\right)\right) = {\frac{{\mathtt{364}}}{{\mathtt{729}}}} = {\mathtt{0.499\: \!314\: \!128\: \!943\: \!758\: \!6}}$$

Melody  Jan 25, 2015
 #1
avatar+92805 
+10
Best Answer

GP

a=1/3     r= 1/3     n=6

 

$$\\\boxed{S_n=\frac{a(1-r^n)}{1-r}}\\\\
S_6=\frac{\frac{1}{3}(1-(\frac{1}{3})^6)}{1-\frac{1}{3}}}\\\\
S_6=\frac{\frac{1}{3}(1-\frac{1}{3^6})}{\frac{2}{3}}}\\\\
S_6=\frac{1}{\not{3}}(1-\frac{1}{3^6})\times \frac{\not{3}}{2}\\\\
S_6=\frac{1}{2}(1-\frac{1}{3^6})\\\\$$

 

$${\mathtt{0.5}}{\mathtt{\,\times\,}}\left({\mathtt{1}}{\mathtt{\,-\,}}\left({\frac{{\mathtt{1}}}{{{\mathtt{3}}}^{{\mathtt{6}}}}}\right)\right) = {\frac{{\mathtt{364}}}{{\mathtt{729}}}} = {\mathtt{0.499\: \!314\: \!128\: \!943\: \!758\: \!6}}$$

Melody  Jan 25, 2015

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