When paying off a loan with initial amount A and monthly payment M at an interes

Question

When paying off a loan with initial amount A and monthly payment M at an interest rate of p percent, the total amount T(n) of the loan after n months is computed by adding p/12 percent to the amount due after n - 1 months and then subtracting the monthly payment M. Convert this description into a recurrence for the amount owed after n months.

Base Formula: T(n) = ( 1 + 0.019/12) * T(n-1) - M

A) Convert this description into a recurrence for the amount owed after n months
B) Prove the Proof

Explanation / Answer

Imports System.Collections.Generic Imports System.Linq Imports System.Text Imports System.Globalization Public Class RecRuleParser Private Sub New() End Sub Public Shared Function ParseRRule(rRule As String, showExceptions As Boolean) As String If rRule = "" Then Return "" End If Dim parsed As String = String.Empty 'set up the return string Dim englishStatement As New StringBuilder() 'Break it into basic parts Dim elements As String() = rRule.Split(vbCrLf) 'it's double spaced in the db, so deal with it accordingly Dim startDate As String = elements(0).Trim Dim endDate As String = elements(1).Trim Dim recRule As String = elements(2).Trim Dim recExcs As String = String.Empty 'check for exceptions If elements.Length > 3 Then recExcs = elements(3).Trim End If 'Attempt to parse the zulu dates into something else Dim dtStart As DateTime = DateTime.ParseExact(getElemValue(startDate), "yyyyMMddTHHmmssZ", System.Globalization.CultureInfo.InvariantCulture, DateTimeStyles.AdjustToUniversal) Dim dtEnd As DateTime = DateTime.ParseExact(getElemValue(endDate), "yyyyMMddTHHmmssZ", System.Globalization.CultureInfo.InvariantCulture, DateTimeStyles.AdjustToUniversal) Dim tsEnd As TimeSpan = dtEnd.Subtract(dtStart) 'Now work with the recurrence rule Dim rruleElems As New Dictionary(Of String, String)() 'Convert the string to a dictionary so we can find things easy parsed = getElemValue(recRule) 'no need having unnecessarily declared strings elements = parsed.Split(";"c) For i As Integer = 0 To elements.Length - 1 Dim tmp As String() = elements(i).Split("="c) rruleElems.Add(tmp(0), tmp(1)) Next englishStatement.Append("Occurs " & rruleElems("FREQ").ToLower()) Dim calType As String = String.Empty 'need a scratchpad 'start translating into English Dim timeToAdd As Integer = 0 Try timeToAdd = Convert.ToInt32(rruleElems("COUNT")) Catch timeToAdd = 0 End Try Dim days As String() Select Case rruleElems("FREQ").ToLower() Case "daily" days = rruleElems("BYDAY").Split(","c) englishStatement.Append(parseDayNames(days)) dtEnd = dtEnd.AddDays(timeToAdd) calType = "days" Exit Select Case "weekly" calType = "weeks" dtEnd = dtEnd.AddDays(timeToAdd * 7) Try days = rruleElems("BYDAY").Split(","c) englishStatement.Append(parseDayNames(days)) Catch 'just in case we missed something on this one Throw New Exception("Error while processing Recurrence Rule") End Try Exit Select Case "monthly" calType = "months" dtEnd = dtEnd.AddMonths(timeToAdd) 'see if it's positional Try Dim bsp As String = getDayEnding(rruleElems("BYSETPOS")) englishStatement.Append(" on the " & bsp & " " & parseDayNames(rruleElems("BYDAY").Split(","c)).Replace(" every ", "")) Catch 'Ok, no BYSETPOS, let's go for BYMONTHDAY Dim bsp As String = getDayEnding(rruleElems("BYMONTHDAY")) englishStatement.Append(" on the " & bsp & " day of each month") End Try Exit Select Case "yearly" calType = "years" dtEnd = dtEnd.AddYears(timeToAdd) 'looks a lot like monthly.... Dim mName As String = CultureInfo.CurrentCulture.DateTimeFormat.GetMonthName(Convert.ToInt32(rruleElems("BYMONTH"))) 'see if it's positional Try Dim bsp As String = getDayEnding(rruleElems("BYSETPOS")) englishStatement.Append(" on the " & bsp & " " & parseDayNames(rruleElems("BYDAY").Split(","c)).Replace(" every ", "") & " of " & mName) Catch 'Ok, no BYSETPOS, let's go for BYMONTHDAY Dim bsp As String = getDayEnding(rruleElems("BYMONTHDAY")) englishStatement.Append(" on the " & bsp & " day of " & mName) End Try Exit Select Case "hourly" calType = "hours" dtEnd = dtEnd.AddHours(timeToAdd) Exit Select Case Else Exit Select End Select englishStatement.Append(" starting on " & dtStart.ToLocalTime().ToShortDateString() & " at " & dtStart.ToLocalTime().ToShortTimeString()) If timeToAdd > 0 Then englishStatement.Append(" for the next " & rruleElems("COUNT") & " " & calType) englishStatement.Append(" ending on " & dtEnd.ToLocalTime().ToShortDateString() & " at " & dtStart.AddHours(tsEnd.Hours).ToLocalTime().ToShortTimeString()) End If If recExcs.Length > 0 AndAlso showExceptions Then Dim excs As String() = recExcs.Split(":"c)(1).Split(","c) Dim retString As String = String.Empty englishStatement.Append(" except on ") For r As Integer = 0 To excs.Length - 1 'we'll use dtEnd, it's not doing anything now dtEnd = DateTime.ParseExact(excs(r), "yyyyMMddTHHmmssZ", System.Globalization.CultureInfo.InvariantCulture, DateTimeStyles.AdjustToUniversal).ToLocalTime() If r 2 Then retString += dtEnd & "," Else If r 1 Then days(d) = " and " & days(d) Else If days.Length > 2 Then days(d) += "," End If End If retString += " " & days(d) Next End If End If Return retString End Function Private Shared Function getDayEnding(d As String) As String 'tried to avoid a big ugly if statement 'handle the events on the "n"th day of the month If d.EndsWith("1") AndAlso d "11" Then d += "st" End If If d.EndsWith("2") AndAlso d "12" Then d += "nd" End If If d.EndsWith("3") AndAlso d "13" Then d += "rd" End If If d.Length < 3 Then 'hasn't been appended yet d += "th" End If Return d End Function End Class

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