The main difference between the two is when these things would end. When you say annuity, this means that the payment will last for a certain period of time. For instance, there would be a specific deadline when the last payment should be sent. When you say perpetuity, this means that the payment is going to go on for an indefinite period of time.
Perpetuity is usually symbolized by the infinity sign. If you would like to compute for this, there are two factors that you need to know. The first one is the payment rate which means that the amount that you are supposed to pay. The second one is the interest rate.
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