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

a.Write out the first five terms of 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 type of sequence described. Since the material loses 50% of its mass every 10 years, this is a geometric sequence where each term is multiplied by a common ratio.

Determine the common ratio \( r \). Because the material retains 50% of its mass each decade, \( r = 0.5 \).

Write the general formula for the \( n^{th} \) term of the sequence: \( M_n = M_0 \times r^n \), where \( M_0 = 20 \) grams is the initial mass.

Calculate the first five terms by substituting \( n = 0, 1, 2, 3, 4 \) into the formula: \( M_0, M_1, M_2, M_3, M_4 \).

Express each term explicitly as \( M_n = 20 \times (0.5)^n \) grams, and write out the values for \( n = 0 \) through \( n = 4 \) without simplifying the numerical results.

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주요 개념

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

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio. In this problem, the mass decreases by 50% every 10 years, so the ratio is 0.5. Understanding geometric sequences helps to write out terms like M₁ = M₀ × 0.5, M₂ = M₁ × 0.5, and so on.

Exponential Decay

Exponential decay describes processes where quantities decrease at a rate proportional to their current value. Radioactive decay is a classic example, where the mass reduces by a fixed percentage over equal time intervals. This concept explains why the mass halves every decade, leading to a geometric sequence.

Sequence Notation and Indexing

Sequence notation uses subscripts to denote terms, such as Mₙ for the nth term. Indexing helps track the progression of the sequence over discrete steps—in this case, decades. Correctly interpreting M₀ as the initial mass and Mₙ as the mass after n decades is essential for writing and understanding the terms.