[tex]\sum _{n=1}^{100}\frac{4}{5^n}\\a_n=\frac{4}{5^n},\:a_{n+1}=\frac{4}{5^{\left(n+1\right)}}\\\:r=\frac{a_{n+1}}{a_n}\\r=\frac{\frac{4}{5^{\left(n+1\right)}}}{\frac{4}{5^n}}=\frac{1}{5}\\a_1=\frac{4}{5}\\r=\frac{1}{5},\:a_n=\frac{4}{5}\left(\frac{1}{5}\right)^{n-1}\\[/tex][tex]\sum _{n=1}^{100}\frac{4}{5^n}=\frac{4}{5}\cdot \frac{1-\left(\frac{1}{5}\right)^{100}}{1-\frac{1}{5}}=\frac{4}{5}\cdot \frac{1-\left(\frac{1}{5}\right)^{100}}{1-\frac{1}{5}}=\frac{5^{100}-1}{5^{100}}\\\\\sum _{n=1}^{100}\frac{4}{5^n}+\frac{1}{5^{100}} \\=\frac{5^{100}-1}{5^{100}}+\frac{1}{5^{100}} =1[/tex]
