Law of total expectation and recursion

I have a problem with one exercise from Probability and Random Processes (2001) by Geoffrey R. Grimmett and David R. Stirzaker.

A coin shows heads with probability $p$. Let $X_n$ be the number of flips required to obtain a run of n consecutive heads. Show that $E(X_n) = \sum\limits_{i=1}^n p^{-i}$.

The solution offered by authors is

$E(X_n) = E\{E(X_n|X_{n-1})\} = E\{p(X_{n-1} + 1) + (1-p)(X_{n-1} + 1 + X_{n})\}$.

I am a bit confused with the last equation.

From definition of $E(X|Y)$ we shoud have something like:$$E(X_n|X_{n-1}) =\begin{cases}v_1, \text{with} \; \text{probability} \; P(X_{n-1} = w_1), \\v_2, \text{with} \; \text{probability} \; P(X_{n-1} = w_2), \\\dots\end{cases}$$But on the right hand we got something like this (from answer to Rigorous proof of the recursion method to compute expectations in probability.)$$E(X_n) = E(X_n|X_n \; \text{ends})P(X_n \; \text{ends}) + E(X_n|X_n \; \text{starts} \; \text{again})P(X_n \; \text{starts} \; \text{again})$$

So could someone expand the given answer in a more clear and strict form.

Excuse my English, I'm not a native speaker.