Calculating gain of an overall system. Analogue electronics
As Dan Mills says, stop thinking of the amplifier system as inverting operational amplifier as the question does not explicitly mention so. Just imagine them to be gain blocks, they could be inverting operational amplifiers but that does not matter. The following is the schematic I came up with the information given.







simulate this circuit &ndash; Schematic created using CircuitLab

Note that there is a voltage divider that halves the output voltage of the first stage to the input of the second stage. Hence the total voltage gain of the system would be 100*0.5*50 = 2500.

For the second part, the question insists that the third stage should be built with a single opamp. So let it be an inverting opamp. We can use an inverting opamp with Rin of 20k as the first stage and the amplifier with output impedence of 2k as the final stage as shown below:







simulate this circuit

To calculate the feedback resistance R2 of the first stage inverting amplifier, we need to use the fact that the overall gain of the system is 2000.

Overall gain = (R/20)*50*(25/25.5)*100=2000.
(Note that voltage divider between the second and third stage)

Therefore, the feedback resistance R2 of the inverting amplifier will be 8.16k.