A perpetuity$-$immediate with effective rate i pays $1,2,3,$dots,$n$ at the end of year $1,2,3,$dots,$n$ and pays n each thereafter. Determine which one of the following is NOT the price.

$($a$)(i|))|)()n$

$($b$)|)$

$($c$)|)+\frac{n}{i}$

$()n$

$($d$)$ None of these

John deposits $100$ into an account at the beginning of each $4-$year period for $40$ years. The account credits interest at an annual effective interest rate of i$.$ Suppose $5A(20)=A(40),$ where $A(20)$ is the accumulation at the end of $20$th year $($right before the next payment$).$ Set up an equation for solving $x^5$ where $x=1+j$ is the $4-$year future value factor.

Smith deposits $x$ each at time $($year$)2,4,$dots,$10$ and $Y$ each at time $1,3,$dots,$9$ into an account that earns effective annual rate i$.$ Find an expression for the total accumulation right after the last deposit and simplify it as much as possible.

Smith deposits $100$ each into an account that earns $12\%$ nominal annual rate compound monthly at the end of each month for $12$ months. He wants to deposit $x$ per month for the next $12$ months so that he'll have exactly $2000$ right after the last deposit. Set up an equation for X$.$

An $n-$year loan of $L$ with effective annual rate i can be paid off with $n$ annual payments of $K$ each at the end of each year. Find an expression of $P{R}_{.}.$

Suppose that the PV of a perpetuity$-$due which pays annual payment of $x$ at end of each year with annual rate of interest $i$ is $5$ times the PV of the same perpetuity right after the $5$th payment. Find an expression for $(1+i).$

$($Bonus$)$ An n$-$payment annuity pays $k$ at time $k=1,2,$dots,$n.$ Show that $|)-n$

$u$ and thus $A(n)=$dots

$($Bonus$)1000$ is deposited into Fund $x,$ which earns an annual effective rate of $6\%.$ At the end