Converting a repeating base $7$ expression to a fraction

Question

A question was given to me to convert a decimal in base seven to a fraction in base seven, where the base $7$ expression was $._7515151515\ldots$. I understand this would be $\frac 57 + \frac 1{49} + \frac 5{343} + \cdots$, but I don't know how to simplify this into the answer, which is $\frac 34$ in base seven.

Answer 1

You want to compute $\sum_{k=0}^\infty \theta {1 \over 49^k}$, where $\theta = {51_7 \over 100_7 } = {36 \over 49 }$.

Since $1+ {1 \over 49} + {1 \over 49^2} + \cdots = {49 \over 48}$, the answer is ${49 \over 48} {36 \over 49 } = { 3 \over 4}$.