= Solution
The time $T$ is the <waiting time for two consecutive successes> in <independent> fair trials. First note its integrability: if any one of the disjoint pairs $(X_1,X_2),\ldots,(X_{2k-1},X_{2k})$ is $++$, then $T\leq2k$. Thus $\mathbb P(T>2k)\leq(3/4)^k$.
Let $e_0$ be the expected additional waiting time with no trailing positive step, and let $e_1$ be the expected additional time when the preceding step was positive. Conditioning on the next <independent> increment gives
$$
e_0=1+\tfrac12e_0+\tfrac12e_1,\qquad e_1=1+\tfrac12e_0.
$$
A negative step returns either state to state $0$; a positive step moves state $0$ to state $1$ and completes the pattern from state $1$. Substitution yields $e_0=2+e_1=3+e_0/2$, so $e_0=6$. The original process starts in state $0$, and therefore
$$
\boxed{\mathbb E T=6.}
$$
Overlapping pairs are dependent, so regarding each successive pair as a fresh trial of success <probability> $1/4$ would give the wrong <expectation>.
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