An annuity pays out from an invested lump sum until it's exhausted; a perpetuity pays forever because you only ever draw the interest. Same machinery as a loan, run in the other direction. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.
A loan and an annuity are the same machine facing opposite directions. Each period the balance is multiplied by (1 + r), and then a payment moves it — down for a loan, up for an investment. Every formula in this topic is that one step repeated, which is why the future value of an annuity turns out to be nothing more exotic than the sum of a geometric series.
An annuity depletes a lump sum by making regular withdrawals that exceed the interest earned, while a perpetuity only withdraws the interest so the principal remains intact forever.
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