Problem

Interpret the base in \(P(t)=800\cdot~1.03^{t}\) as a recurrence factor and retained percentage, determine the yearly growth rate, verify the interpretation with \(P(0)\) and \(P(1)\), and distinguish the coefficient.

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An exponential base is the factor multiplying the current amount during each input interval. We’ll express that action as a recurrence, convert the factor to a retained percentage, and separate the original whole from the growth portion. Comparing consecutive modeled values will verify the factor while keeping it distinct from the initial coefficient.

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base = 1.03; recurrence = P(t+1)=1.03P(t); retained = 103%; growth = 103%/year; check = P(1)=824; coefficient distinction = 800 is initial population

base = 800; recurrence = P(t+1)=800P(t); retained = 800%; growth = 799%/year; check = P(1)=824; coefficient distinction = 1.03 is initial population

base = 1.03; recurrence = P(t+1)=P(t)+1.03; retained = 103%; growth = 3%/year; check = P(1)=801.03; coefficient distinction = 800 is initial population

base = 1.03; recurrence = P(t+1)=1.03P(t); retained = 103%; growth = 3%/year; check = P(0)=800 and P(1)=824; coefficient distinction = 800 is initial population

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Interpret the base in \(P(t)=800\cdot~1.03^{t}\) as a recurrence factor and retained percentage, determine the yearly growth rate, verify the interpretation with \(P(0)\) and \(P(1)\), and distinguish the coefficient.

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