Question

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the 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 \(M_{n}\) be the mass of the radioactive material at the end of the \(n\) th decade, where the initial mass of the material is \(M_{0}=20 \mathrm{g}\)

Short answer

Answer: The explicit formula for the mass sequence is \(M_n = M_0 * (\frac{1}{2})^n\). The estimated limit of the mass sequence is 0.

Step-by-step solution

Write out the first five terms of the sequence.

The problem states that the mass of the radioactive material at the end of the \(n\) th decade has decayed by \(50 \%\) compared to the previous decade. This means that the mass halves with each passing decade. We are given that \(M_0 = 20\) g, so we can compute \(M_1, M_2, M_3, M_4, M_5\) as follows: - \(M_1 = M_0 * \frac{1}{2} = 20 * \frac{1}{2} = 10\) g - \(M_2 = M_1 * \frac{1}{2} = 10 * \frac{1}{2} = 5\) g - \(M_3 = M_2 * \frac{1}{2} = 5 * \frac{1}{2} = 2.5\) g - \(M_4 = M_3 * \frac{1}{2} = 2.5 * \frac{1}{2} = 1.25\) g - \(M_5 = M_4 * \frac{1}{2} = 1.25 * \frac{1}{2} = 0.625\) g

Find an explicit formula for the terms of the sequence.

Since the mass of the radioactive material at the end of the \(n\) th decade has decayed by \(50 \%\) compared to the previous decade, we can find an explicit formula for the mass sequence as follows: \(M_n = M_0 * (\frac{1}{2})^n\)

Find a recurrence relation that generates the sequence.

A recurrence relation that generates the mass sequence can be expressed as: \(M_{n+1} = M_n * \frac{1}{2}\), with the initial condition \(M_0 = 20\) g

Estimate the limit of the mass sequence.

As the mass at any decade is decreasing and approaching zero, we can estimate the limit of the mass sequence using the explicit formula: \(\lim_{n \to \infty} M_n = \lim_{n \to \infty} 20 * (\frac{1}{2})^n = 0\) Since the mass is always positive and decreasing, the limit of the mass sequence is 0.

Key concepts

Sequence and Series

In understanding radioactive decay, it's crucial to grasp the concept of a sequence and a series. A sequence is an ordered list of numbers following a particular rule. In our case, the sequence represents the amount of radioactive material remaining after each decade. Series, on the other hand, involve the sum of the elements in a sequence, which might not be directly applicable to the radioactive decay scenario but is still a fundamental concept in mathematics.

For example, the sequence generated by the radioactive decay is as follows: 20g, 10g, 5g, 2.5g, 1.25g, and so on. Each term is half the previous one, illustrating a systematic reduction over time. Understanding sequences and series is foundational to many areas of mathematics and science.

Explicit Formula

An explicit formula allows us to find any term in a sequence without needing to know the previous terms. It's a direct way to calculate the nth term. The explicit formula for our radioactive decay problem is given by:\[M_n = M_0 \times \left(\frac{1}{2}\right)^n\]In this case, if you want to find out how much mass remains after 60 years (or 6 decades), you simply plug 6 into the formula to get \(M_6 = 20 \times \left(\frac{1}{2}\right)^6\). This hands-on approach makes the explicit formula a powerful tool in understanding and predicting sequence behaviors, like radioactive decay.

Recurrence Relation

A recurrence relation defines each term of a sequence using its preceding terms. In our decay problem, the recurrent relation is:\[M_{n+1} = M_n \times \frac{1}{2}\]This means that to find a specific term, you start from the initial value and apply the relation repeatedly. The recurrence relation emphasizes the process over time, which can be more intuitive when considering dynamic systems such as radioactive decay.

Limits of a Sequence

Understanding the concept of limits is important when we're looking at processes that continue indefinitely. In sequences, a limit describes the value that the sequence approaches as the number of terms goes to infinity. For the radioactive decay sequence we're discussing, the limit is:\[\lim_{n \to \infty} M_n = 0\]This tells us that as time progresses, the amount of radioactive material will approach zero, indicating that it will eventually decay to an undetectable amount. While we'll never actually reach a mass of 0g, it becomes negligibly small over time.

Exponential Decay

Radioactive decay is a classic example of exponential decay, a process where a quantity decreases at a rate proportional to its current value. This creates a rapidly decreasing curve when plotted on a graph. With half of the material decaying every 10 years, we see that the mass of our radioactive material diminishes exponentially over time. It's a continuous process characterized by the formula:\[M(t) = M_0e^{-kt}\]where \(M_0\) is the initial mass, \(e\) is the base of the natural logarithm, \(k\) is the decay constant, and \(t\) represents time. In our simplified example, the decay is modeled in discrete intervals (decades), but the concept of exponential decay extents to continuous situations as well.