Expected tosses for a run of ten heads

A fair coin is tossed repeatedly until it has shown heads $10$ times in a row. What is the expected total number of tosses?

$2046$. Let $E_k$ be the expected tosses needed for a run of $k$ heads. To get $k$ in a row you first need $k-1$ in a row, then one more toss; half the time that toss is a tail and you restart, so $E_k = 2E_{k-1}+2$. With $E_0=0$ this gives $E_k = 2^{k+1}-2$, and $E_{10}=2^{11}-2=2046$.