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


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


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.

Verified step by step guidance

1. Identify the type of sequence described. Since the material loses 50% of its mass every 10 years, the sequence \( M_n \) represents a geometric sequence where each term is half the previous term.
2. Write the general formula for the sequence. Given the initial mass \( M_0 = 20 \) grams and the decay factor of 0.5 per decade, the mass at the end of the \( n^{th} \) decade is given by: \[ M_n = M_0 \times (0.5)^n \]
3. Understand what the limit of the sequence represents. The limit \( \lim_{n \to \infty} M_n \) corresponds to the mass remaining after infinitely many decades, which models the long-term behavior of the radioactive material.
4. Use a calculator or graphing utility to evaluate the terms of the sequence for large values of \( n \) to observe the trend. This will help estimate the limit by seeing how \( M_n \) behaves as \( n \) increases.
5. Based on the geometric sequence properties and the decay factor, conclude whether the limit exists and what its value approaches as \( n \to \infty \).

Key Concepts

Sequences and Their Limits

A sequence is an ordered list of numbers generated by a specific rule. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding limits helps determine the long-term behavior of sequences, such as whether the mass stabilizes or diminishes to zero in decay problems.

Exponential Decay

Exponential decay describes processes where a quantity decreases by a fixed percentage over equal time intervals. In radioactive decay, the mass reduces by a constant factor (here, 50%) every decade, modeled by Mₙ = M₀ * (decay factor)ⁿ. This concept is essential for setting up the sequence formula.

Use of Technology for Estimation

Calculators and graphing utilities can approximate sequence values and their limits when analytical solutions are complex or to verify results. They help visualize the sequence behavior over many terms, making it easier to estimate the limit or conclude if it does not exist.