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

$S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\cdots ...\left(i\right)$

$\frac{S}{7}=0+\frac{2}{7}+\frac{6}{7^2}+\frac{12}{7^3}+\frac{20}{7^4}+\cdots ...\left(ii\right)$

On subtracting the equation $\left(ii\right)$ from equation $\left(i\right)$, we get

$\frac{6S}{7}=2+\frac{4}{7}+\frac{6}{7^2}+\frac{8}{7^3}+\cdots ...\left(iii\right)$

$\frac{6S}{49}=0+\frac{2}{7}+\frac{4}{7^2}+\frac{6}{7^3}+\cdots ...\left(iv\right)$

On subtracting the equation $\left(iv\right)$ from  equation $\left(iii\right)$ , we get

$\left(\frac{6}{7}-\frac{6}{49}\right)S=2+\frac{2}{7}+\frac{2}{7^2}+\frac{2}{7^3}+\cdots$

$\frac{36}{49}S=2\left(1+\frac{1}{7}+\frac{1}{7^2}+\cdots \infty \right)$

$\Rightarrow S=2·\frac{1}{1-\frac{1}{7}}·\frac{49}{36}$

$\Rightarrow 4S={\left(\frac{7}{3}\right)}^3$