a, [TEX]S= \frac{1}{7}+ \frac{1}{7^2}+ \frac{1}{7^3}+...+ \frac{1}{7^{100}}[/TEX]
\Rightarrow [TEX] \frac{1}{7}S= \frac{1}{7^2}+ \frac{1}{7^3}+...+ \frac{1}{7^{101}}[/TEX]
\Rightarrow [TEX]S-\frac{1}{7}S= \frac{1}{7}+ \frac{1}{7^2}+ \frac{1}{7^3}+...+ \frac{1}{7^{100}}- \frac{1}{7^2}- \frac{1}{7^3}-...- \frac{1}{7^{101}}[/TEX]
\Rightarrow [TEX]\frac{6}{7}S= \frac{1}{7}- \frac{1}{7^{101}}[/TEX]
\Rightarrow [TEX]\frac{6}{7}S=\frac{7^{100}-1}{7^{101}}[/TEX]
\Rightarrow [TEX]S= \frac{7^{100}-1}{7^{101}}:\frac{6}{7}=\frac{7^{100}-1}{7^{100}.6} [/TEX]
b,[TEX]S= \frac{4}{5}+ \frac{4}{5^2}+ \frac{4}{5^3}+...+ \frac{4}{5^{100}}[/TEX]
\Rightarrow [TEX]5S=4+ \frac{4}{5}+ \frac{4}{5^2}+ \frac{4}{5^3}+...+ \frac{4}{5^{99}}[/TEX]
\Rightarrow[TEX]5S-S= 4+ \frac{4}{5}+ \frac{4}{5^2}+ \frac{4}{5^3}+...+ \frac{4}{5^{99}}-\frac{4}{5}- \frac{4}{5^2}- \frac{4}{5^3}-...- \frac{4}{5^{100}}[/TEX]
\Rightarrow[TEX]S= (4- \frac{4}{5^{100}}):4[/TEX]
\Rightarrow[TEX]S=1- \frac{1}{5^{100}} [/TEX]