Question

 

B =  1/7 + 1/7²  + 1/7³   + ………. + 1/7¹⁰⁰    

 

 

Answer 1

\(B=\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+...+\frac{1}{7^{100}}\)

\(7B=1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}\)

\(7B-B=\left(1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}\right)-\left(\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+...+\frac{1}{7^{100}}\right)\)

\(6B=1-\frac{1}{7^{100}}\)

\(B=\frac{1-\frac{1}{7^{100}}}{6}\)

Answer 2

\(B=\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{100}}\)

\(\Rightarrow7B=1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}\)

\(\Rightarrow7B-B=\left(1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}\right)-\left(\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{100}}\right)\)

\(\Rightarrow6B=1-\frac{1}{7^{99}}\)

\(\Rightarrow B=\left(1-\frac{1}{7^{99}}\right):6\)

Answer 3

\(\Leftrightarrow7B=1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}\)

\(\Leftrightarrow7B-B=1-\frac{1}{7^{100}}\)

\(\Leftrightarrow B=\frac{1-\frac{1}{7^{100}}}{6}\)