72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence

c.Find a recurrence relation that generates the sequence.

Radioactive decay
A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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Identify the nature of the sequence: Since the material loses 50% of its mass every 10 years, the mass at the end of each decade is half of the mass at the previous decade. This indicates a geometric sequence where each term is multiplied by a constant ratio.

Define the recurrence relation: The mass at the end of the (n+1)ᵗʰ decade, denoted as \(M_{n+1}\), depends on the mass at the end of the nᵗʰ decade, \(M_n\). Since the mass halves every decade, the relation can be written as \(M_{n+1} = r \times M_n\), where \(r\) is the common ratio.

Determine the common ratio \(r\): Because the material transmutes 50% of its mass, the remaining mass is 50%, so \(r = 0.5\).

Write the full recurrence relation: Substitute the value of \(r\) into the relation to get \(M_{n+1} = 0.5 \times M_n\).

Include the initial condition: The problem states the initial mass is \(M_0 = 20\) grams, which is necessary to generate the sequence starting point.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Recurrence Relations

A recurrence relation defines each term of a sequence using one or more previous terms. It provides a way to generate the sequence step-by-step, often reflecting a process evolving over discrete intervals, such as time periods in decay problems.

Exponential Decay

Exponential decay describes a process where a quantity decreases by a consistent percentage over equal time intervals. In radioactive decay, the mass reduces by a fixed fraction (e.g., 50%) every period, leading to a sequence where each term is a constant fraction of the previous one.

Initial Conditions in Sequences

Initial conditions specify the starting value(s) of a sequence, such as M₀ = 20g here. They are essential for uniquely determining the sequence generated by a recurrence relation and for calculating subsequent terms accurately.