simple interest

Suppose a bank account is opened at time $0$ and $M_0$ is deposited into the account. A simple interest is interest with the following characteristics:

1. it is earned at subsequent time periods $t,2t,\mathrm{\dots }$, where $t$ is the length of the initial time interval (1 for 1 month, 12 for 1 year, etc…)

2. the interest earned at the end of each time period is the same regardless of the time period

The following table illustrates the structure of the simple interest.

time period at	principal	interest	interest accrued
$0$	$M_0$	$0$	$0$
$t$	$M_0$	$i$	$i$
$2t$	$M_0$	$i$	$2i$
$3t$	$M_0$	$i$	$3i$
$\mathrm{⋮}$	$\mathrm{⋮}$	$\mathrm{⋮}$	$\mathrm{⋮}$
$nt$	$M_0$	$i$	$ni$

$The\mathrm{`}\mathrm{`}tota{l}^{\prime \prime }interest$i(nt)$earned(accrued)attheendoftime$nt$is$ni$.Iftheaccountisclosedandthemoneywithdrawnattheendof$nt$,andthetotalamountofmoneyreceivedis$$M(nt)=M_0+ni.$$Theinterestrateassociatedwiththesimpleinterestaspresentedabovebetweentwotimeperiods,say$at$and$bt$,isgivenby$$r(at,bt)=\frac{1}{M_0}\frac{i(bt)-i(at)}{bt-at}=\frac{i}{M_{0}t},$$whichdoesnotdependonthechoiceof$a$and$b$.Inotherwords,theoriginalprincipal$M_0$,theamountofinterest$i$,andthelengthoftheinitialtimeinterval$t