Which appeal is John Adams using in this letter to a fellow colonist who
supported the king?

The correct answer to this open question is the following.

The appeal is John Adams using in this letter to a fellow colonist who supported the king was logos, the appeal to logic or reason that is supported by arguments.

John Adams tried to convince those loyalists to change their minds and support the Revolutionary movement. He used logic because he stated the reasons and gave arguments. He literary wrote: "That a representation in parliament is impracticable, we all agree; but the consequence is, that we must have a representation in our supreme legislatures here. This was the consequence that was drawn by kings, ministers, our ancestors, and the whole nation, more than a century ago, when the colonies were first settled."

Let's remember the other two rhetoric devices: Pathos and Ethos. Pathos is the use of emotion to appeal to audiences. Ethos is the use of the experience or reputation of the speaker to convince an audience.  

twelve to twenty-four months