To determine the present value of an annuity due, 1 is added to the discount factor of the ordinary annuity. However, to determine the future value of an annuity due, 1 is removed from the discount factor of the ordinary annuity.
What is the mathematical principle and/or intuition for this difference? I understand that broadly, PV and FV are inverses/opposites of one another but I need to know what exactly is going.