Array's cumulative sum

If you want to achieve a cumulative array array like Array(a,a+b,a+b+c) from Array(a,b,c), then this is the function to achieve it, if you want to pass start and end parameters:

Public Sub TestMe()

    Dim outputArray     As Variant
    Dim inputArray      As Variant
    Dim counter         As Long

    inputArray = Array(1, 2, 4, 8, 16, 32, 64)

    outputArray = generateCumulativeArray(inputArray, 1, 4)
    For counter = LBound(outputArray) To UBound(outputArray)
        Debug.Print outputArray(counter)
    Next counter

    outputArray = generateCumulativeArray(inputArray, toValue:=4)
    For counter = LBound(outputArray) To UBound(outputArray)
        Debug.Print outputArray(counter)
    Next counter

End Sub

Public Function generateCumulativeArray(dataInput As Variant, _
                                        Optional fromValue As Long = 0, _
                                        Optional toValue As Long = 0) As Variant

    Dim i                   As Long
    Dim dataReturn          As Variant
    ReDim dataReturn(0)
    dataReturn(0) = dataInput(fromValue)

    For i = 1 To toValue - fromValue
        ReDim Preserve dataReturn(i)
        dataReturn(i) = dataReturn(i - 1) + dataInput(fromValue + i)
    Next i
    generateCumulativeArray = dataReturn
End Function

Concerning just summing an array, this is the way to do it: You can use the WorksheetFunction. and you can pass the array as an argument. Thus, you get all the functions, e.g. Average, Min, Max etc:

Option Explicit

Public Sub TestMe()

    Dim k As Variant
    k = Array(2, 10, 200)
    Debug.Print WorksheetFunction.Sum(k)
    Debug.Print WorksheetFunction.Average(k)

End Sub

If you want the sum from a given start to a given end, the easiest way is probably to make a new array and to sum it completely. In Python this is called slicing, in VBA this could be done a bit manually:

Public Sub TestMe()

    Dim varArr          As Variant
    Dim colSample       As New Collection

    varArr = Array(1, 2, 4, 8, 16, 32, 64)

    colSample.Add (1)
    colSample.Add (2)
    colSample.Add (4)
    colSample.Add (8)

    Debug.Print WorksheetFunction.Sum(generateArray(varArr, 2, 4))
    Debug.Print WorksheetFunction.Sum(generateArray(colSample, 2, 4))

End Sub

Public Function generateArray(data As Variant, _
                              fromValue As Long, _
                              toValue As Long) As Variant

    Dim i                   As Long
    Dim dataInternal        As Variant
    Dim size                As Long

    size = toValue - fromValue
    ReDim dataInternal(size)

    For i = LBound(dataInternal) To UBound(dataInternal)
        dataInternal(i) = data(i + fromValue)
    Next i

    generateArray = dataInternal

End Function

The idea is that the generateArray function returns a new array. Thus, its complete sum is what you need. It works also with collections, not only with arrays. Be careful, when using collections, they start with index 1, while arrays (usually) start with 0. If you want to use the same indexing for Arrays and Collections, then change the generateArray function to this one:

Public Function generateArray(data As Variant, _
                              fromValue As Long, _
                              toValue As Long) As Variant

    Dim i                   As Long
    Dim dataInternal        As Variant
    Dim size                As Long

    size = toValue - fromValue
    ReDim dataInternal(size)

    If IsArray(data) Then
        For i = LBound(dataInternal) To UBound(dataInternal)
            dataInternal(i) = data(i + fromValue)
        Next i
    Else
        For i = LBound(dataInternal) To UBound(dataInternal)
            dataInternal(i) = data(i + fromValue + 1)
        Next i
    End If

    generateArray = dataInternal

End Function

Or write Option Base 1 on top and the array will start from 1 (not advised!).

For cumulative sum try the following

Function CumulativeSum(Data() As Integer, k As Integer) As Integer
    Dim tempArr
    tempArr = Data
    ReDim Preserve temp(0 To k - 1)
    CumulativeSum = WorksheetFunction.Sum(tempArr)
End Function

EDIT :

Sub Demo()
    Dim MyArray
    Dim i As Long
    MyArray = Array(1, 2, 3, 4, 5, 6, 7, 8, 9)

    Debug.Print MyArray(LBound(MyArray))
    For i = LBound(MyArray) + 1 To UBound(MyArray)
        MyArray(i) = MyArray(i - 1) + MyArray(i)
        Debug.Print MyArray(i)
    Next i
End Sub

Above code updates array arr from
1, 2, 3, 4, 5, 6, 7, 8, 9
to
1, 3, 6, 10, 15, 21, 28, 36, 45

Try this:

Sub test()
Dim arr As Variant
arr = Array(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

Dim mySum As Long, k As Long
Dim wsf As WorksheetFunction
Set wsf = Application.WorksheetFunction

k = 6
'operative line below
mySum = wsf.Sum(wsf.Index(arr, 1, Evaluate("ROW(1:" & k & ")")))

MsgBox mySum

End Sub

This function returns an array with the cumulative sum of the original vector.

Function CumuVector(Vec As Variant) As Variant()
Dim element, v() As Variant
Dim i As Integer
lastindexinvec = 0
For Each element In Vec
    lastindexinvec = last + 1
Next 
ReDim v(lastindexinvec) As Variant
i = 0
For Each element In Vec
    If i < last Then
    sum = sum + element
    v(i) = sum
    i = i + 1
    End If
Next
CumuVector = v
End Function