A battery with internal resistance r is connected to a load resistance R. If R is increased, does the terminal voltage of the battery increase, decrease, or stay the same?

The terminal voltage of the battery decreases.

Explanation:

An idea battery does not have internal resistance but a real practical battery always have an internal resistance r connected in series with the battery.

This internal resistance causes a voltage drop when load R is connected, a current I flows in the circuit which causes a voltage drop I*r in the battery therefore, the terminal voltage of the battery will decrease.