Question

M=1/3+1/9+1/27+...+1/2187

Answer 1

`#Huvo`

`M=1/3+1/9+1/27+...+1/2187`

`M xx 3 = 3 xx (1/3+1/9+1/27+1/81+1/243+1/729+1/2187)`

`M xx 3 = 1 + (1/3+1/9+1/27+1/81+1/243+1/729)`

`M xx 3 = 1+M-1/2187`

`M xx 2 = 1 - 1/2187`

`M xx 2 =2186/2187`

`M= 2186/2187 : 2`

`M=1093/2187 `

Answer 2

\(\dfrac{1}{3}M=\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^8}\)

\(\Leftrightarrow-\dfrac{2}{3}M=\dfrac{1}{3^8}-\dfrac{1}{3}\)

\(\Leftrightarrow M=\dfrac{1-3^7}{3^8}:\dfrac{-2}{3}=\dfrac{1-3^7}{3^8}\cdot\dfrac{3}{-2}=\dfrac{3^7-1}{2\cdot3^7}\)