Expected flips for two heads in a row

A fair coin is flipped until two heads appear in a row ( $HH$ ). What is the expected number of flips?

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Track two states: no progress ( $a$ ) and one head so far ( $b$ ). Write $a$ and $b$ via one flip.
$a = 1 + \tfrac12 b + \tfrac12 a$ and $b = 1 + \tfrac12 a$ ; solve for $a$ .

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