a)derive the likelihood estimator of the geometric function

let represent number of attacks,,

b)determine the cramer-rao lower bound for an unbiased estimator of

I have some answers for part a) which gave the MLE as which would work for part b) as it is unbiased...for it to be unbiased the proof is that either MSE(estimate)=var(estimate) or BIAS(estimate)=0 implies E(estimate)=0. hence

however I cannot see how this estimator was obtained.

My working is as follows andI obtained something completely different while the official working gave the anser above: any offerings as to how they did it?

which is clearly biased