# Unbiased sufficient statistic for $1/p$ of geometric distribution

Suppose that a random variable $$X$$ has the geometric distribution with unknown parameter $$p$$, where the geometric probability mass function is: $$f(x;p)= p(1-p)^x,\qquad x=0,1,2,\ldots ; \quad 0 Find a sufficient statistic $$T(X)$$ that will be an unbiased estimator of $$1/p$$.

Now I know the population mean is $$\frac{1-p}{p}$$ and the sufficient statistic for p is a function of $$\sum_{i=1}^{n}X_i$$. But I am unsure on how to proceed any help greatly appreciated!

Assuming $$X_1,X_2,\ldots,X_n$$ are i.i.d with pmf $$f$$.
You have $$E_p(X_1+1)=\frac1p$$ for all $$p\in(0,1)$$, so
$$E_p\left[\frac1n\sum_{i=1}^n (X_i+1)\right]=E_p\left[\frac1n\sum\limits_{i=1}^n X_i+1\right]=\frac1p\quad,\,\forall\,p\in(0,1)$$