Rolling until you match the first roll

EasyProbability Expected Value~4m

You roll a fair six-sided die once and remember the value. You then keep rolling the die until you get a value at least as large as that first roll. Let $N$ be the number of rolls after the first. What is $\mathbb{E}[N]$ ?

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Condition on the first roll $k$ . On later rolls, what is the probability of landing $\ge k$ ?
Given that success probability, the wait is geometric with mean $1/p_k$ . Average $6/(7-k)$ over $k=1,\dots,6$ .

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2.45 (± 0.01)

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Rolling until you match the first roll

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