Bring the berries, sugar and 1 cup of water to a boil and cook, smashing the fruit, for about 5 minutes or until the back of a spoon can be coated. Essentially flavored simple syrup. Strain. Toss the pulp or save for something else.
Make the limeade by mixing together the syrup made above, the lime juice and 4 cups of water.
To make the Adult Arnold Palmer, mix equal parts of your limeade made above, cooled green (or black) tea, and vodka. Other additions that would be nice is a splash of soda water or lemon/lime soda. If it isn't sweet enough, you can add more simple syrup.