A $20-$year continuous stream of payments consists of payments at a rate of $3,000$ per year for the first $10$ years, then at a rate of $2,000$ per year from $t=10$ to $t=20.$ At an interest rate of $6\%$ convertible monthly, what is the present value of this payment stream?

$2.$

A $10-$year annuity$-$immediate has a first$-$year payment of $500.$ The subsequent payments increase by $100$ each year. Find the present value of this annuity at an annual effective interest rate of $5\%.$

$3.$

& A $5-$year annuity$-$immediate has a first$-$year payment of $1,000.$ The subsequent payments decrease by $100$ each year. Find the present value of this amnuity at an annual effective interest rate of $6\%.$

$4.$

An annuity pays $1$ at the end of each of the next four years and $2$ at the end of each of the four following years. Based on a $5\%$ annual effective interest rate, what is the present value of this annuity?

$5.$

$2$ An annuity pays $100$ at the end of each of the next $10$ years and $200$ at the end of each of the five subsequent years. If $i=0\mathrm{.}08,$ find the present value of the annuity.

$6.$

$8$ An annuity$-$immediate has a first payment of $100,$ and the payments increase by $100$ each year until they reach $500.$ There are $10$ further payments of $500.$ Find the present value of this annuity at $6\mathrm{.}5\%.$