I started studying Z-transforms this week and was not sure about the below question.

Work out the first 4 coefficients of the impulse response of the discrete time filter, which is characterized by a zero at +1 and a pole at -0.5 using only the power-series method.

__Question__Work out the first 4 coefficients of the impulse response of the discrete time filter, which is characterized by a zero at +1 and a pole at -0.5 using only the power-series method.

I have attempted to work out the solution but I am not certain about my answer and the initial Z transform function that is used. Please can someone check my answer and method of working.

thank you all.