This should be rather standard so I hope somebody with a good background in probability theory would give me a quick solution or a reference.

We are given a threshold positive integer $T>0$. Let $a\_1=1$ and for all $k$ with probability one half set $a\_k=3a\_{k-1}$ or else $a\_k=2a\_{k-1}$. We will stop the process at smallest time $\tau$ when $a\_{\tau} \geq T$. We would like to compute the constant $c$ defined to be,

$ E[ \sum\_{i=1}^{\tau} a\_i ] = c T + o(T) $

Could you estimate $c$ ?