Expected tosses for runs of one, two, three heads

A fair coin is tossed repeatedly. Let $X_k$ be the number of tosses needed until $k$ heads in a row first appear. Find $\mathbb{E}(X_1)$, $\mathbb{E}(X_2)$ and $\mathbb{E}(X_3)$.

$2,\ 6,\ 14$. To extend a run of $k-1$ heads you need one more toss; a tail resets you, giving $E_k=E_{k-1}+1+\tfrac12E_k$, i.e. $E_k=2E_{k-1}+2$. Starting from $E_0=0$ this yields $2^{k+1}-2$.