Expected number of tosses to get 3 consecutive heads, what's wrong with my solution? Apparently the answer is 14 Expected number of tosses to get 3 consecutive Heads, but I got 11. Can someone pinpoint the error to my solution?
Let $X_i$ = Expected number of tosses to ith head. Hence,
$$X_1 = 1 + 1/2*X_1$$
$$X_2 = X_1 + 1/2 + 1/2*X_2$$
$$X_3 = X_2 + 1/2 + 1/2*X_3$$
I know the first line is correct. Regarding the 2nd and 3rd lines, my reasoning is that for the $i$th roll, it's expected value must be the $(i-1)$th roll plus 50% chance of the next role being heads and if not, then everything is reset back to 0. 
So, the Expected value of getting 2 heads is the expected value of getting one head + 50% chance of immediately getting the second head + the expected value of getting a tail, which resets the state back to the beginning.
-Edit-
Just to clarify since there is some confusion, I let $X_1, X_2, X_3$ to represent the expected value getting to 1,2, and 3 consecutive heads, respectively.

Your argument is almost correct.
Consider $X_2$
Your type of reasoning means that half the time this will be $X_1+1$ and half the time this will be $(X_1+1)+X_2$. So your second line should have "+1+" instead of "+1/2+". 
Ditto in line 3.
Then your method gives $X_1=2, X_2=6,X_3=14.$
