Maths alevel A2 help. Geometric sequence to recurrence equation

Does anyone know how to convert from geometric sequences to recurrence relationship, for example: Un = 3((2/3)^n) - 1

Is this in the alevel spec for OCR mei as I cant find how to do it in the book but there was a questions on it on the edxecel textbook which i use.

Its not that bad and you should know that a simple recurrence relationship like
U_{n+1} = r U_n
represents a geometric sequence with common ratio r and U_0 = a for the initial value?

So here its ~a geometric sequence so one way is to ivevsimply sub in n+1 instead of n
U_{n+1} = 3 (2/3)^(n+1) - 1
and try and get the right hand side in terms of "3((2/3)^n) - 1" which can be replaced by U_n. Alternatively, start with U_n and try and do a couple of simple transformations to get the expression which gives U_{n+1}

ive gotten up to= 2(2/3)^n + 1, but then how do i go to the answer which is: (2Un - 1)/2

this is what the mark scheme says however im just confused about how they go from the last line to the answer

Its wrong - should be over 3. Factor out 2/3 as per the previous post and just work it through and post what you get if necessary. Fairly obviously, if you forgot about the constant, the common ratio between successive terms is 2/3, hence the denominanotr should be 3.

yes, i see thats where i was going wrong as I thought the textbook was right, thankyou for your help.