Problem

Day \(1\) starts with a \(40~\text{mg}\) medicine dose, and the dose halves each day. A student proposes \(a_{n}~=~40(1/2)^{n}\) for \(n~=~1,~2,~3,~\ldots~\). Compare the required and proposed starting values, diagnose the indexing error, and write the corrected rule.

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The required start is a₀ = 40 mg and the proposal gives a₀ = 40 mg, so there is no error; aₙ = 40(1/2)ⁿ for n ≥ 0.

The required start is a₁ = 40 mg and the proposal gives a₁ = 20 mg, but there is no error; aₙ = 40(1/2)ⁿ for n ≥ 1.

The required start is a₁ = 40 mg, but the proposal gives a₁ = 20 mg. The decay factor is wrong; aₙ = 40(1/4)ⁿ⁻¹ for n ≥ 1.

The required start is a₁ = 40 mg, but the proposal gives a₁ = 20 mg. Halving was applied one day too early; aₙ = 40(1/2)ⁿ⁻¹ for n = 1, 2, 3, …

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