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Let $S = \frac{1}{7} + \frac{2}{7^2} + \frac{3}{7^3} + \frac{1}{7^4} + \frac{2}{7^5} + \frac{3}{7^6}$.
\n" ); document.write( "We can group the terms in sets of three:
\n" ); document.write( "$S = \left(\frac{1}{7} + \frac{2}{7^2} + \frac{3}{7^3}\right) + \left(\frac{1}{7^4} + \frac{2}{7^5} + \frac{3}{7^6}\right)$.
\n" ); document.write( "Let $A = \frac{1}{7} + \frac{2}{7^2} + \frac{3}{7^3}$.
\n" ); document.write( "Then $S = A + \frac{1}{7^3} A = A\left(1 + \frac{1}{7^3}\right)$.
\n" ); document.write( "We have $A = \frac{1}{7} + \frac{2}{49} + \frac{3}{343} = \frac{49}{343} + \frac{14}{343} + \frac{3}{343} = \frac{49+14+3}{343} = \frac{66}{343}$.
\n" ); document.write( "Then $S = \frac{66}{343}\left(1 + \frac{1}{343}\right) = \frac{66}{343}\left(\frac{343+1}{343}\right) = \frac{66}{343}\left(\frac{344}{343}\right) = \frac{66\cdot 344}{343\cdot 343} = \frac{22704}{117649}$.\r
\n" ); document.write( "\n" ); document.write( "We can also write
\n" ); document.write( "$S = \sum_{k=0}^1 \left( \frac{1}{7^{3k+1}} + \frac{2}{7^{3k+2}} + \frac{3}{7^{3k+3}} \right) = \sum_{k=0}^1 \frac{1 \cdot 7^2 + 2 \cdot 7 + 3}{7^{3k+3}} = \sum_{k=0}^1 \frac{49+14+3}{7^{3k+3}} = \sum_{k=0}^1 \frac{66}{7^{3k+3}}$.
\n" ); document.write( "$S = \frac{66}{7^3} + \frac{66}{7^6} = \frac{66}{343} + \frac{66}{117649} = \frac{66\cdot 343 + 66}{117649} = \frac{66(343+1)}{117649} = \frac{66(344)}{117649} = \frac{22704}{117649}$.\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{\frac{22704}{117649}}$
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