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Chapter 1: Problem 20

20\. $$a_{1}=3, a_{n+1}=\sqrt{3 a_{n}}$$

Short Answer

Expert verified
Every term in the sequence is 3; \(a_n = 3\) for all \(n\).

Step by step solution

01

Understand the Sequence

We are dealing with a recursively defined sequence where the first term \(a_1 = 3\) and each subsequent term is defined by \(a_{n+1} = \sqrt{3 a_n}\). This means to find \(a_2\), \(a_3\), and so on, we will use the last found term and calculate the next one using this formula.
02

Calculate the Second Term

Substitute \(n = 1\) into the recursive formula to find \(a_2\): \[a_2 = \sqrt{3a_1} = \sqrt{3 \times 3} = \sqrt{9} = 3.\] Thus, \(a_2 = 3\).
03

Calculate the Third Term

Similarly, substitute \(n = 2\) into the recursive formula to find \(a_3\): \[a_3 = \sqrt{3a_2} = \sqrt{3 \times 3} = \sqrt{9} = 3.\] Thus, \(a_3 = 3\).
04

General Observations for the Sequence

From steps 2 and 3, we can observe that \(a_2\) and \(a_3\) are both 3, just like \(a_1\). As actually verified by calculation, it appears each term \(a_n\) equals 3, suggesting \(a_{n+1} = 3\). Therefore, by the recursion given, all terms in the sequence are equal to 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus for Life Sciences
Calculus is a fundamental aspect of life sciences, offering tools to model and analyze biological phenomena. In the life sciences, calculus helps in understanding growth models, population dynamics, and much more. Consider a situation where you're observing a population whose size changes over time, similar to a sequence.
  • Population Growth: Calculus can aid in analyzing how a population evolves by modeling it with sequences and differential equations.
  • Decaying Substances: In the study of medication within the body, calculus comes into play to describe how drugs decrease in concentration over time.
  • Temperature Fluctuations: Calculating how an organism's internal temperature responds to external variations can involve sequences and series.
In our sequence example, we're essentially observing repetitive, unchanging growth. Understanding the broader applications of such sequences in real-life biological and environmental contexts enriches your grasp of calculus's role in life sciences.
Sequence Recursion
Sequence recursion is a fascinating concept in mathematics, allowing us to define sequences with a rule that relates each term to previous ones. Unlike sequences defined outright, recursive sequences build gradually.In our exercise, we have a recursive sequence starting with a base of 3, and each subsequent term depends on the previous one through the formula:
\(a_{n+1} = \sqrt{3a_n}\).
  • Defining the Base: Every recursive sequence begins with an initial term, here \(a_1 = 3\). This starting point is crucial as it dictates subsequent values.
  • Recursive Formula: The rule applied to get from one term to the next,
    guides the formation of the sequence. It's a function of the preceding terms.
  • Structure of the Sequence: The repeated application of the formula results in an interesting pattern.
    For this sequence, every term remains constant at 3.
Recursive sequences can become more complex with varied rules, but they offer a systematic approach to generating terms.
Mathematical Induction
Mathematical induction is a powerful tool, mainly used to prove that a statement is true for every natural number. The concept is akin to falling dominoes; if you push the first, the rest will follow.
  • Base Case: Induction starts by proving the validity of a statement for an initial case, often involving the smallest number.
  • Inductive Step: The next part is assuming the statement holds for an arbitrary number \(n\).
    Then you show it must also work for \(n+1\), confirming a crucial chain of logic.
In terms of sequences, like our example, induction could show that if \(a_1 = 3\), then all subsequent terms must also equal 3. This is because each term relies entirely on its predecessor. Should the initial terms comply with the pattern, induction confirms this continues indefinitely. This method provides certainty in mathematical statements with repeated logical links.

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Most popular questions from this chapter

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If \(f\) and \(g\) are both even functions, is \(f+g\) even? If \(f\) and \(g\) are both odd functions, is \(f+g\) odd? What if \(f\) is even and \(g\) is odd? Justify your answers.
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