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

b.Find an explicit formula for the 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, the sequence represents exponential decay, which is a geometric sequence.

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

Write the general form of the explicit formula for a geometric sequence: \( M_n = M_0 \times r^n \), where \( M_0 \) is the initial mass and \( n \) is the number of decades.

Substitute the given initial mass \( M_0 = 20 \) grams and the common ratio \( r = 0.5 \) into the formula to get \( M_n = 20 \times (0.5)^n \).

Interpret the formula: \( M_n \) gives the mass of the radioactive material after \( n \) decades, showing how the mass decreases by half every 10 years.

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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 an explicit formula for the nth term based on the initial value and the common ratio.

Explicit Formula for Sequences

An explicit formula expresses the nth term of a sequence directly in terms of n, without needing previous terms. For geometric sequences, the formula is Mₙ = M₀ * rⁿ, where M₀ is the initial term and r is the common ratio. This allows calculation of any term in the sequence efficiently.

Radioactive Decay Modeling

Radioactive decay describes how a substance decreases over time at a consistent fractional rate. Modeling decay with sequences involves applying the decay factor repeatedly over equal time intervals. This real-world context helps connect abstract sequence concepts to practical applications.