$[10$ points$]$ Barbie make $$2,000$ deposits at the end of each year to Bank A with

an effective annual rate of $4\%$ for $20$ years starting this year. After $20$ years, Barbie stops

depositing into the bank but keeps the balance in the bank which continues to earn the same

interest rate for $5$ more years.

Ken also make $$2,000$ deposits at the end of each year to Bank B with an effective annual

rate of $6\%$ for $20$ years. However, Ken's first deposit is deferred $5$ years, i$.$e$.,$ made at the

end of year $6.$

$($a$)$ Calculate the present value at $t=0$ of Barbie's deposits.

$($b$)$ Calculate the accumulated value at the end of the $25$th year of Barbie's deposits.

$($c$)$ Calculate the present value at $t=0$ of Ken's deposits.

$($d$)$ Calculate the accumulated value at the end of the $25$th year of Ken's deposits.