un = u0*rn for n = 1,2,3, ...
where r is the constant multiple.
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I expect that you mean the formula for calculating the terms in a geometric sequence. Please see the link.
what is the recursive formula for this geometric sequence?
Yes, that's what a geometric sequence is about.
A geometric sequence is : a•r^n while a quadratic sequence is a• n^2 + b•n + c So the answer is no, unless we are talking about an infinite sequence of zeros which strictly speaking is both a geometric and a quadratic sequence.
It is called arithmetico-geometric sequence. I have added a link with some nice information about them.
A single number does not constitute a sequence.