Official solution
\[S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \] ..... ...... (i)
\[{1 \over 7}S = {2 \over 7} + {6 \over {{7^2}}} + {{12} \over {{7^3}}} + {{20} \over {{7^4}}} + \] .... ....... (ii)
(i) - (ii)
\[{6 \over 7}S = 2 + {4 \over 7} + {6 \over {{7^2}}} + {8 \over {{7^3}}} + \] ...... ....... (iii)
\[{6 \over {{7^2}}}S = {2 \over 7} + {4 \over {{7^2}}} + {6 \over {{7^3}}} + \] ..... ......... (iv)
(iii) - (iv)
\[{\left( {{6 \over 7}} \right)^2}S = 2 + {2 \over 7} + {2 \over {{7^2}}} + {2 \over {{7^3}}} + \] ......
\[ = 2\left[ {{1 \over {1 - {1 \over 7}}}} \right] = 2\left( {{7 \over 6}} \right)\]
\[\therefore\] \[4S = 8{\left( {{7 \over 6}} \right)^3} = {\left( {{7 \over 3}} \right)^3}\]