state 0 : initial state
state 1 : H
state 2 : HT
state 3 : HTT (final state)
Initially you're in state 0. You will remain in state 0 until you flip a H (in which case you'll move to state 1).
Once in state 1, you will remain there until you flip a T (in which case you'll will move to state 2).
And once in step 2, you'll move to state 3 if you flip a T, and move back to state 1 if you flip a H.
Now let be the expected amount of flips in takes to reach state 3 starting from state i.
You can set up three equations by conditioning on your first flip after you enter a state.
now solve for
I get 8 flips.