one of the most striking phi facts is that a series formed by adding the two previous numbers, such as 2 , 3 , 5 , 8 , 13 , 21… always ends up with
no matter which two numbers are used for the first two terms.
Here is an excel visual basic demonstration. The test sub asks the user for two integers. The series is computed, the 20th term is divided by the 19th term, and the result displayed to the screen. Phi is subtracted from the result to show the difference. All 20 terms and the results are also written to the debug window.
Sub test_phi_converge() Dim num1 As Long Dim num2 As Long num1 = InputBox("enter first integer:") num2 = InputBox("enter second integer:") Call phi_converge(num1, num2) End Sub Sub phi_converge(numa As Long, numb As Long) Dim numc As Long Dim i As Integer Debug.Print numa Debug.Print numb 'this will make a series of 20 'and divide term_20 by term_19 For i = 3 To 19 numc = numa + numb 'c becomes b and b becomes a for the next iteration numa = numb numb = numc Debug.Print numc Next i 'term_20 numc = numa + numb Debug.Print numc Dim result As Double, diff As Double result = numc / numb diff = result - 1.61803398874989 MsgBox result & " " & Format(diff, "#0.00000000000000") Debug.Print result Debug.Print Format(diff, "#0.00000000000000") End Sub
a sample run