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.

$The\mathrm{`}\mathrm{`}tota{l}^{\prime \prime }interest$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