I am currently studying for the actuarial exam FM with the mathematical interest theory textbook from MAA. I have been doing a bunch of problems and the one below is giving me a lot of trouble to figure out. I have tried defining the nominal discount rate in terms of the annual effective rate trying to solve for d. This effort has been fruitless. I stated the problem below two times just to clarify. Any hint that can point me to the direction of solving it will be appreciated.
Given that the nominal discount rate compounded semi-annually minus the annual effective discount rate = .00107584, find the nominal interest rate compounded tri-annually minus the annual effective interest rate.
Given $d^{(2)}-d=.00107584$, find $i^{(3)}-i$
note: $$\left(1-\frac{d^{(2)}}{2}\right)^2=1-d$$ and since you know $d^{(2)}-d=.00107584\Rightarrow d^{(2)}=d+.00107584,$ substitute and you will end up with a quadratic in $d$. Once you know $d$, you can use equalities like the one above to convert $d$ to $i^{(3)}$ and $i$. $$\left(1-\frac{d^{(m)}}{m}\right)^{-m}=(1-d)^{-1}=1+i=\left(1+\frac{i^{(n)}}{n}\right)^n$$ FM is a LOT of disguised quadratics and conversions. Know your identities. It is not as rich a site, but the quantitative finance stack site is much more helpful for aspiring actuaries since the material is all finance.
EDIT: I changed the statement of equality at the bottom to reflect variable $m$'thly and $n$'thly rates since this is the general case.