How about we consider an and r to be the initial term and the normal proportion of the G.P. separately.

Given, \[\mathbf{a}\text{ }=\text{ }\mathbf{1}\]

\[\mathbf{a3}\text{ }=\text{ }\mathbf{ar2}\text{ }=\text{ }\mathbf{r2}\]

\[\mathbf{a5}\text{ }=\text{ }\mathbf{ar4}\text{ }=\text{ }\mathbf{r4}\]

Then, at that point, from the inquiry we have

\[\mathbf{r2}\text{ }+\text{ }\mathbf{r4}\text{ }=\text{ }\mathbf{90}\]

\[\mathbf{r4}\text{ }+\text{ }\mathbf{r2}\text{ }\text{ }\mathbf{90}\text{ }=\text{ }\mathbf{0}\]

Thus, the normal proportion of the G.P. is \[\pm \mathbf{3}.\]