At an effective annual interest rate of $i,i>0,$ both of the following annuities have a present value of $x.$

$($a$)$ a $20$ year annuity$-$immediate with annual payments of $$50.($b$)$ a $30$ year annuity immediate that pays $$52\mathrm{.}50$ for the first $10$ years, $$40$ for the next $10$ years,

and then pays $$10$ for the last $10$ years.

Find $x.$