Let S = 2 + frac{6}{7} + frac{12}{7^{2}} + fr | vedclass

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(C) Given $S = 2 + \frac{6}{7} + \frac{12}{7^{2}} + \frac{20}{7^{3}} + \frac{30}{7^{4}} + \ldots$ $(1)$Divide by $7$: $\frac{S}{7} = \frac{2}{7} + \fr

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$\left(\frac{7}{3}\right)^{3}$

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(C) Given $S = 2 + \frac{6}{7} + \frac{12}{7^{2}} + \frac{20}{7^{3}} + \frac{30}{7^{4}} + \ldots$ $(1)$Divide by $7$: $\frac{S}{7} = \frac{2}{7} + \frac{6}{7^{2}} + \frac{12}{7^{3}} + \frac{20}{7^{4}} + \ldots$ $(2)$Subtracting $(2)$ from $(1)$:$S - \frac{S}{7} = 2 + \left(\frac{6-2}{7}ight) + \left(\frac{12-6}{7^{2}}ight) + \left(\frac{20-12}{7^{3}}ight) + \ldots$$\frac{6S}{7} = 2 + \frac{4}{7} + \frac{6}{7^{2}} + \frac{8}{7^{3}} + \ldots$ $(3)$Divide $(3)$ by $7$: $\frac{6S}{49} = \frac{2}{7} + \frac{4}{7^{2}} + \frac{6}{7^{3}} + \ldots$ $(4)$Subtracting $(4)$ from $(3)$:$\frac{6S}{7} - \frac{6S}{49} = 2 + \left(\frac{4-2}{7}ight) + \left(\frac{6-4}{7^{2}}ight) + \left(\frac{8-6}{7^{3}}ight) + \ldots$$\frac{42S - 6S}{49} = 2 + \frac{2}{7} + \frac{2}{7^{2}} + \frac{2}{7^{3}} + \ldots$$\frac{36S}{49} = 2 + \left(\frac{2/7}{1 - 1/7}ight) = 2 + \left(\frac{2/7}{6/7}ight) = 2 + \frac{1}{3} = \frac{7}{3}$$S = \frac{7}{3} 	imes \frac{49}{36} = \frac{343}{108}$$4S = 4 	imes \frac{343}{108} = \frac{343}{27} = \left(\frac{7}{3}ight)^{3}$

Let $S = 2 + \frac{6}{7} + \frac{12}{7^{2}} + \frac{20}{7^{3}} + \frac{30}{7^{4}} + \ldots$. Then $4S$ is equal to

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