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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 $(X_1,X_2),\dots ,(X_{2k-1},X_{2k})$ is $++$, then $T\le 2k$. Thus $\mathbb{P}(T>2k)\le (3/4{)}^k$.

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 givesA 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 thereforeOverlapping pairs are dependent, so regarding each successive pair as a fresh trial of success probability $1/4$ would give the wrong expectation.