An annuity has end of quarter payments for fifteen years, and the pay$-$ment at the end of the j$-$th quarter $($j $=1,2,...,60)$ is $$100$j$.$The payments are made directly into a savings account with a nominal in$-$terest rate of $6\%$ payable monthly, and they are left in the account.

$($a$)$ Find the effective interest rate for a quarter, and use it to com$-$pute the balance in the savings account immediately after the last payment.

$($b$)$ Use Fact $(4\mathrm{.}4\mathrm{.}3)$ to recalculate the balance in the savings account immediately after the last payment. Make sure that your answer agrees with your answer to part $($a$),$ and note which method you found easier. Use financial math techniques, no excel. the answer is $251,477\mathrm{.}70$