Let $X$ be geometric random variable with parameter $p$.
How to prove that:
(1) $E[X-1|X>1] = E[X]$
(2) $E[X^2|X>1] = E[(X+1)^2]$
Author explains the fact below and states it is used to prove (1)
$$P(X-1=k|X>k) = P(X=k).$$
I understood how the fact is true but could not understand how it is used to derive (1)
(2) was stated without explanation. Could someone help in deriving (1) and (2)?
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