what i know :  if i want to find  $Nth$ term of a fibonacci series like :   1   1   2   3   5   8   13  21 .......   then to find $6th$ term we use golden ratio like:    so it becomes like :     well and good.now  what i want to know :  what changes should i make in golden ratio or while applying it for different fibonacci series . For example :   1   6   7   13   20   33 .......   now if i want to find the 10th no. of above series then what changes should i do in golden ratio... now what is the value of X10's R.H.S(right hand side)  X10 = ? (for this fibonacci series)
DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic.  QUESTION: What is the significance of starting the Fibonacci sequence with $0,1$ ?  For instance, if I picked any two random integers, say 2 and 7, to start a sequence would I actually be creating some multiple or derivation of the Fibonacci sequence?  Is there a general mathematical explanation for the relationship between any sequence represented by $a[0] = x, a[1] = y, a[n] = a[n-1] + a[n-2]$ and the Fibonacci sequence?  Or, back to my example sequence, is there a general mathematical relationship between:  $2,7,9,16,25,41,66,107,173,280...$  and  $0,1,1,2,3,5,8,13,21,34...$  Perhaps the Golden Ratio explains it somehow?  Any help would be appreciated.
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