halp!

The common ratio of the series is $b \over 7$, and the starting term is $b \over 7$, so we have the equation: $\large{{{b \over 7} \over 1 - {b \over 7}} = 7}$

Multiplying the entire equation by $1 - {b \over 7}$ gives us ${b \over 7} = 7(1 - {b \over 7})$, which simplifies to ${b \over 7} = 7 - b$. 

Multiplying the entire equation by 7 gives us $b = 49 - 7b$, meaning $b = \color{brown}\boxed{49 \over 8}$