Let \({u_1},\;{u_1}r\) and \({u_1}{r^2}\) be the three consecutive terms of the geometric sequence.If \({u_1}\) and \({u_1}r\) are the first and fourth terms, respectively, of an arithmetic sequence then \({u_1}r - {u_1} = 3d\); and if \({u_1}r\) and \({u_1}{r^2}\) are the first and eighth terms, respectively, then \({u_1}{r^2} - {u_1} = 7d\).Solving for \(d\) in each of these equations gives:\({u_1}r - {u_1} = 3d\;\;\; \Rightarrow \;\;\;d = \frac{1}{3}\left( {{u_1}r - {u_1}} \right)\);and \({u_1}{r^2} - {u_1} = 7d\;\;\; \Rightarrow \;\;\;d = \frac{1}{7}\left( {{u_1}{r^2} - {u_1}} \right)\)\(d = \frac{1}{3}\left( {{u_1}r - {u_1}} \right) = \frac{1}{7}\left( {{u_1}{r^2} - {u_1}} \right)\)\(d = 7\left( {{u_1}r - {u_1}} \right) = 3\left( {{u_1}{r^2} - {u_1}} \right)\;\;\; \Rightarrow \;\;\;7{u_1}r - 7{u_1} = 3{u_1}{r^2} - 3{u_1}\)