Question

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.

Accepted Answer

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. 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:

$$ M_n = M_0 	imes (0.5)^n $$ Understand what the limit of the sequence represents. The limit $$ \lim_{n 	o \infty} M_n $$ corresponds to the mass remaining after infinitely many decades, which models the long-term behavior of the radioactive material. 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. Based on the geometric sequence properties and the decay factor, conclude whether the limit exists and what its value approaches as $$ n 	o \infty .$$

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