On May $9$th $2024,$ Arthur opens an investment account at Elwood City Bank, and immediately deposits a lump sum $($initial deposit$).$ He continues to make regular deposits thereafter. $($a$)$ Suppose that Arthur$$s initial deposit is $$2000.$ He makes regular deposits on $5$th of each month. Interest compounds

on the $1$st and $20$th of each month, at an APR of $10\%.$ arthur would like the account balance to reach $$20,000$ on

January $15$th $2026.$

Determine how much money Arthur should deposit regularly so that he achieves his goal. Round your answer

to two decimal places. $($b$)$Suppose that Arthur makes no initial deposit. He makes regular deposits of $$1,000$ on the $5$th of each month. Interest

compounds on the last day of each month at an APR of $6\%.$ Arthur would like the account balance to eventually

reach $$40,000.$

Determine the month and year on which Arthur will achieve his goal. Your answer must be in the form $$Month$,$

Year.$($Bonus point if the correct day is given, and the final answer is presented in the form $$Month$,$ Day, Year.$)$