The common difference of the A.P. $$\frac{1}{{2b}},$$ $$\frac{{1 - 6b}}{{2b}},$$  $$\frac{{1 - 12b}}{{2b}},$$   . . . . . is

$$\eqalign{ & A.P.\,is\,\frac{1}{{2b}},\,\frac{{1 - 6b}}{{2b}},\,\frac{{1 - 12b}}{{2b}},.... \cr & \Rightarrow \frac{1}{{2b}},\,\frac{1}{{2b}} - \frac{{6b}}{{2b}},\,\frac{1}{{2b}} - \frac{{12b}}{{2b}},\,.... \cr & \Rightarrow \frac{1}{{2b}},\,\frac{1}{{2b}} - 3,\,\frac{1}{{2b}} - 6,\,.... \cr & \therefore d = \frac{1}{{2b}} - 3 - \frac{1}{{2b}} = - 3 \cr} $$