poojadholam
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A more complete expression for this situation is:
Vout = 1.65[1/(1+(1+Rf/Rn)/A)] - dVi(Rf/Rn)
where dVin is your 1uV above 1.65
Yes, if we call Vadd the added voltage to the input of Rn with respect to Vin (that is the voltage applied to the non inverting pin), then
Vout = Vin - Rf/Rn*Vadd
Ideally if Vadd=0 then Vout=Vin regardless the gain.
I don't understand why you call this configuration "VGA" that stands for "Variable Gain Amplifier".
I agree with confusion regarding the terminology. You haven't explained what's "Variable" about the VGA, are you varying the resistors? Gain of 1000 is a lot and will take care in many respects to get it to perform well.
As to your specific question I wouldn't be surprised at all by the result your seeing. Offsets can easily explain that. Keep testing it and see if it performs in the other ways you expect.