Can someone please help explain the logic and reasoning behind the formula for calculating the future value of an annuity?

Example:

If \$10,000 is invested at the end of each year for the next 10 years, earning 10% compounded annually, how much money will be in the account at the end of the 10th year?

I understand that if I calculated it manually, it would look like this to give the resulting answer \$159,374.25

Year 0: 0 (because this is the present value, no payment yet, payment is at end of first year)

Year 1: 0 * 1.1 + 10,000 = 10,000

Year 2: 10,000 * 1.1 + 10,000 = 21,000

Year 3: 21,000 * 1.1 + 10,000 = 33,100

Year 4: 33,100 * 1.1 + 10,000 = 46,410

Year 5: 46,410 * 1.1 + 10,000 = 61,051

Year 6: 61,051 * 1.1 + 10,000 = 77,156.1

Year 7: 77,156.1 * 1.1 + 10,000 = 94,871.71

Year 8: 94,871.71 * 1.1 + 10,000 = 114,358.88

Year 9: 114,358.88 * 1.1 + 10,000 = 135,794.77

Year 10: 135,794.77 * 1.1 + 10,000 = 159,374.25

Question:

How does that manual calculation relate to this formula?
Future Value = (payment per period / rate per period) * [(1 + rate per period) ^ number of periods - 1]

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