Problem 56160. Ordinary Annuity : Rate (Present Value)

Created by Mehmet OZC

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<div style = "text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; "><div style="block-size: 230.377px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.997px 115.188px; transform-origin: 406.997px 115.188px; vertical-align: baseline; "><div style="block-size: 21.0018px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.996px 10.5009px; text-align: left; transform-origin: 383.996px 10.5009px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Theorem : Present Value of an Ordinary Annuity</span></span></div><div style="block-size: 37.6379px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.996px 18.8189px; text-align: left; transform-origin: 383.996px 18.8189px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="116" height="37.5" style="width: 116px; height: 37.5px;"></span></div><div style="block-size: 21.0018px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.996px 10.5009px; text-align: left; transform-origin: 383.996px 10.5009px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">where;</span></span></div><ul style="block-size: 81.7555px; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 390.993px 40.8778px; transform-origin: 390.998px 40.8778px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 20.4389px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 362.994px 10.2137px; text-align: left; transform-origin: 363px 10.2194px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">P : Present value of all payments</span></span></li><li style="block-size: 20.4389px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 362.994px 10.2137px; text-align: left; transform-origin: 363px 10.2194px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">A : Periodic payment</span></span></li><li style="block-size: 20.4389px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 362.994px 10.2137px; text-align: left; transform-origin: 363px 10.2194px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">r : Interest rate per period</span></span></li><li style="block-size: 20.4389px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 362.994px 10.2137px; text-align: left; transform-origin: 363px 10.2194px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">n : Number of periods (peyments)</span></span></li></ul><div style="block-size: 21.0018px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.996px 10.5009px; text-align: left; transform-origin: 383.996px 10.5009px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Calculate r, given P, A and n. Assume interest rate is in [0, 1].</span></span></div></div></div>

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Problem 56160. Ordinary Annuity : Rate (Present Value)

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