Which of the following two sentences is correct?
There is one primary station and multiple secondary stations.
There are one primary station and multiple secondary stations.
I think the first one is correct. But I don't know the reason.
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The first one is correct. Often we have the verb match the closest subject, and elide the other.
So the bold here would be a correct sentence, and the brackets contain the implied verb:
There is one primary station and [there are] multiple secondary stations.
There is one over here and [there is] one over there.
There are three stations nearby and [there is] one further away.