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[TEX] \blue A=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+[/tex][tex] \blue \frac{1}{5^{100}} [/tex]

[tex] \blue 5A= 1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{99}} \\ 5A-A=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{99}}-\frac{1}{5}-\frac{1}{5^2}-\frac{1}{5^3}-...-\frac{1}{5^{100}} \\ 4A=1-\frac{1}{5^{100}} \\ A=\frac{1-\frac{1}{5^{100}}}{4} [/TEX]
 
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