If the increasing annuity$-$immediate accumulated value $(Is{)}_3=6\mathrm{.}5,$ find an exact expression for the rate of interest per period for $i.$

A$.\frac{3\sqrt[\phantom{2}]{2}-5}{7}$

B$.\frac{\sqrt[\phantom{2}]{53}-7}{4}$

C$.\frac{\sqrt[\phantom{2}]{23}-3}{2}$

D$.\frac{3\sqrt[\phantom{2}]{2}-4}{2}$

E$.$ Cannot be determined from the given information.