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Chapter 11: Problem 56

Find the first 40 terms of the sequence defined by $$ a_{n+1}=\left\\{\begin{array}{ll}{\frac{1}{2} a_{n}} & {\text { if } a_{n} \text { is an even number }} \\ {3 a_{n}+1} & {\text { if } a_{n} \text { is an odd number }}\end{array}\right. $$ and \(a_{1}=11 .\) Do the same if \(a_{1}=25 .\) Make a conjecture about this type of sequence.

Short Answer

Expert verified
Both sequences enter cycles, resembling Collatz-like behavior.

Step by step solution

01

Understanding the Sequence Rules

We have a sequence where each term is determined based on the parity (odd or even nature) of the previous term (\(a_n\)). If \(a_n\) is even, the next term \(a_{n+1}\) is \(\frac{1}{2} a_n\). If \(a_n\) is odd, the next term is \(3a_n + 1\). We will calculate the first 40 terms for \(a_1 = 11\) and then for \(a_1 = 25\).
02

Calculating the Sequence from \(a_1 = 11\)

Starting with \(a_1 = 11\), which is odd, apply the rule \(3a_1 + 1 = 34\). Then, \(a_2 = 34\) (even), apply \(\frac{1}{2}\times 34 = 17\). Continue this process, applying the rules based on the parity of each subsequent term until you reach 40 terms.
03

List the Sequence for \(a_1 = 11\)

Following the rules:1. \(11\) (odd), \(3(11) + 1 = 34\)2. \(34\) (even), \(\frac{34}{2} = 17\)3. \(17\) (odd), \(3(17) + 1 = 52\)4. \(52\) (even), \(\frac{52}{2} = 26\)... continue this until you reach 40 terms. Note the sequence follows a cycle observed after calculations, dependent on starting conditions.
04

Calculating the Sequence from \(a_1 = 25\)

Starting with \(a_1 = 25\), which is odd, apply the rule \(3a_1 + 1 = 76\). Then, \(a_2 = 76\) (even), apply \(\frac{1}{2} \times 76 = 38\). Continue applying the rules based on the parity of each next term until 40 terms are found.
05

List the Sequence for \(a_1 = 25\)

Following the rules:1. \(25\) (odd), \(3(25) + 1 = 76\)2. \(76\) (even), \(\frac{76}{2} = 38\)3. \(38\) (even), \(\frac{38}{2} = 19\)4. \(19\) (odd), \(3(19) + 1 = 58\)... continue listing until 40 terms are achieved. Patterns may emerge and cycle similar to known sequences.
06

Observing Patterns and Making a Conjecture

Both sequences generated show oscillating behavior where they eventually fall into a known pattern if the terms are examined further into a larger series, resembling parts of the Collatz conjecture. Based on both starting points, sequences seem to lead towards cycles depending on starting values.

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Key Concepts

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

Parity
Parity refers to whether a number is odd or even, a concept used heavily in mathematics and computer science. In terms of sequences like the one here, parity dictates how each number, or term, in the sequence is transformed to yield the next term.
In our exercise:
  • If a number is even, it's divisible by 2. The sequence applies the rule of dividing by 2 to generate the next number.
  • If the number is odd, it's not divisible by 2. Here, the sequence applies the rule of multiplying by 3 and then adding 1 to proceed.
Understanding parity is essential, as it determines the direct rule application that affects the sequence progression. Parity makes these transformations predictable, depending on whether you start with an even or an odd number.
Sequence
A sequence in mathematics is an ordered list of numbers that follow a specific pattern. In this problem, the sequence alters based on the Collatz Conjecture, a famous mathematical sequence predictor.
Starting with an initial term, each subsequent term is determined by a formula that relies on the parity of the current term:
  • Even numbers lead to terms halved from the current one.
  • Odd numbers generate terms by multiplying the current one by 3 and adding 1.
This leads to a fascinating sequence cycle where particular patterns may emerge, often resembling previous parts of the sequence, and sometimes the sequence reaches known cycles or repeating sequences.
Iterations
Iterations are the repetitive application of a process or a rule. For our sequence exercise, each transformation of a term to the next is an iteration.
You apply the sequence's rule based on the parity repeatedly:
  • Take the current term,
  • Determine its parity,
  • Apply the respective rule (either dividing by 2 or multiplying by 3 then adding 1),
  • Record the new term, and repeat.
Iterations define how sequences evolve and build over time. Performing multiple iterations eventually reveals patterns or cycles that might suggest a conjecture about the sequence's behavior like the Collatz Conjecture suggests.
Even and Odd Numbers
Even and odd numbers play a pivotal role in determining the progression of sequences. They allow us to define a clear, rule-based transformation for each entry in the sequence.
Key characteristics:
  • Even numbers: divisible by 2, often resulting in a smaller next term when the rule \({1\over 2}a_n\) is applied.
  • Odd numbers: not divisible by 2, which generally result in larger next terms, calculated using the rule \((3a_n + 1)\).
Recognizing the starting number's parity鈥攚hether a number is even or odd鈥攄etermines the sequence's direction and how rapidly it reaches a stable series or pattern.

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Most popular questions from this chapter

(a) Expand 1\(/ \sqrt[4]{1+x}\) as a power series. (b) Use part (a) to estimate 1\(/ \sqrt[4]{1.1}\) correct to three decimal places.

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty} \frac{(n !)^{n}}{n^{4 n}}$$

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=\sqrt{x}, \quad a=4, \quad n=2, \quad 4 \leqslant x \leqslant 4.2$$

Use a computer algebra system to find the Taylor polynomials \(T_{n}\) centered at \(a\) for \(n=2,3,4,5 .\) Then graph these polynomials and \(f\) on the same screen. $$f(x)=\sqrt[3]{1+x^{2}}, \quad a=0$$

\(27-30\) Use a power series to approximate the definite integral to six decimal places.\ $$\int_{0}^{0.2} \frac{1}{1+x^{5}} d x$$
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