Show proper working using pen and paper please. Show all steps and explanations.

Dan wins the Iowa Lottery for a grand prize of $$Z.$ A cash payment is not an option. Rather than receiving a cash payment of $$Z,$ Dan is offered two payout options:

$($i$)$ A $20-$year annuity due with $$18,391\mathrm{.}42$ annual level payments.

$($ii$)$ A $10-$year deferred perpetuity$-$due with $$16,718\mathrm{.}57$ annual level payments.

The present value of both options equals $$Z$ at an annual effective rate of $i\%.$ Dan chooses option II as his payout option.

Dan is anxious to receive all of his winnings immediately. So$,$ he makes a deal with his bank to receive a single payment immediately for an amount $$Y.$ The bank calculated $$Y$ using an effective annual interest rate of $(i+1)\%.$