Determine which of the following is FALSE.

$($a$)$ The buying power of $1+r$ one year later is the same as $1$ now.

$($b$)$ The inflation adjusted one year growth factor is $\frac{1+i}{1+r}$

$($c$)$ The inflation adjusted rate of interest is $\frac{i-r}{1+r}$

$(d)$ The inflation adjusted rate $of$ interest $is\frac{1-r}{i+r}(e)$ none $of$ these

$50,000$ each is deposited at the end of years $1,2,$dots,$17.x$ each is withdrawn at the beginning of $18$ th$,19$ th$,20$ th and $21$ st years to make the balance $0.$ If $i$ is the effective annual interest rate, which of the following can be used to solve for $x?$

$($a$)]$

$()i()i[()i]$

$()i[()i$

$($c$)]$

$[()3($b$)]$

$[()i$

$($e$)$ none of these

$($d$)]$

$()i()i[()i]$

$()i[()i$

An account is compounded annually. For a fraction of a year the simple interest $i$ is used. If the account is closed and reopened immediately every month during a year, then the annual rate of return is

$($a$)i$

$($b$)(\frac{i}{12}{)}^{12}$

$($c$)(1+\frac{i}{12}{)}^{12}-1$

$($d$)(1+\frac{i}{12}{)}^{\frac{1}{12}}-1$

$($e$)$ none of these

One account pays interest $i^{(4)}=8\%$ and the other pays interest at a force of $.$ If the two accounts start with the same $A(0)$ and ended with the same $A(5),$ then

$($a$)=ln(1\mathrm{.}04)$

$($b$)=4ln(1\mathrm{.}02)$

$($c$)=ln(1\mathrm{.}02)$

$($d$)=2ln(1\mathrm{.}04)$

$($e$)$ none of these