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Chapter 5: Problem 7

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be the integer sequence defined recursively by 1) \(a_{1}=0 ;\) and, 2) For \(n>1, a_{n}=1+a_{(n / 2)}\). Prove that \(a_{n}=\left[\log _{2} n\right]\) for all \(n \in \mathbf{Z}^{+}\).

Short Answer

Expert verified
By using mathematical induction which consists of two steps namely the base step which successfully proved that the statement holds for initial \(n\), and the induction step where we assumed it's true for any \(n=k\) and proved for \(n=k+1\), we verify that \(a_n = \left[ \log_2 n \right]\) for all positive integers.

Step by step solution

01

Understand Basis of Induction

Mathematical induction has two steps: the base case, and the inductive step. The base step, or the basis of induction, proves the statement for the initial value, which in this case is \(n=1\). Plug \(n=1\) in equation i.e. \(a_1\). From the exercise, we know \(a_1 = 0\). Now plug \(n=1\) in \(a_n = \left[\log_2 n\right]\) i.e. \(a_1 = \left[\log_2 1\right]\) which gives us 0. So, this verifies that the base case is true.
02

Assume for \(n=k\)

The second part of mathematical induction is assumption or hypothesis step. We want to show it's true for \(n=k\). For some positive integer \(k\), we assume that the equation \(a_k = \left[\log_2 k\right]\) is true. This is called the hypothesis of induction.
03

Prove for \(n=k+1\)

Now we would like to prove the statement for the next integer \(n=k+1\). According to recursive relation \(a_{k+1}=1+a_{(k+1) / 2}\). If \(k+1\) is even then we can write it as \(a_{k+1}=1+a_{k/2}\). If \(k+1\) is odd, then we should look at \(k+1=2m+1\) for some integer \(m\) and try proving \(a_{2m+1} \leq 1 + \left[ \log_2 m \right] \leq \left[ \log_2 (2m+1) \right]\).
04

Completing the Induction process

Using the hypothesis, we have \(1 + a_{k/2} = 1 + \left[ \log_2(k/2) \right] = \left[ \log_2{(k+1)} \right]\) for \(k+1\) is even. Observe if \(2m < 2m+1 < 2m+2 \Rightarrow m < 2m+1 < 2(m+1)\). If we take \(\log_2\) of these inequalities, we get \(\log_2{m} < \log_2(2m+1) < \log_2(2) + \log_2(m+1)\), thus \(a_{2m+1} \leq 1 + \left[ \log_2(m) \right] = \left[ \log_2{(2m+1)} \right]\). Therefore, both cases of \(n=k+1\) being odd and even are proved.

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Key Concepts

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

Recursive Sequences
A recursive sequence is a sequence of numbers where each term is defined based on the previous terms. These sequences rely on a starting value, often called the base case, and a rule to define the subsequent terms. In our exercise, the sequence begins with \(a_1 = 0\).
For any \(n > 1\), the sequence is defined as \(a_n = 1 + a_{(n/2)}\).
Recursive sequences are important because they allow the definition of complex sequences with relatively simple rules.
  • The base case provides the starting point.
  • The recursive rule offers a way to find all other terms.
Understanding the recursive sequence in this problem helps us decode how each term relates back to our base case, thus preparing us to prove statements about the terms using mathematical induction.
Inductive Hypothesis
The inductive hypothesis is a central part of mathematical induction. In this process, you assume the statement to be true for a particular \(k\), known as the inductive step, and then use this assumption to prove the statement for \(k+1\).
In the problem, we assume \(a_k = \left[\log_2 k\right]\).
This assumption serves as the foundation for proving the next stage, \(n = k + 1\).
  • You accept the statement as true for \(k\), giving a solid base for further exploration.
  • Ultimately, the goal is to transition smoothly from one step to the next by proving that if the statement holds for \(k\), it should hold for \(k+1\) as well.
The hypothesis provides a logical framework, ensuring that each step of induction builds appropriately on the last.
Integer Sequence
An integer sequence consists of a sequence of numbers, where each number is an integer. The sequence discussed in this problem starts with \(a_1\) and follows the recursive rule provided.
Understanding integer sequences is crucial as they are widely used in various mathematical contexts.
These sequences can grow quickly, so having a formula or recursive rule helps manage and predict the terms.
  • In recursion, sequences allow us to define this integer sequence compactly.
  • They serve as a reference for proving characteristics or properties about the numbers in the sequence.
Working with integer sequences involves predicting integer outcomes, fostering skills in mathematical reasoning and logic.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions and are a crucial tool for solving equations where the variable is in an exponent.
The notation \(\log_2 n\) indicates a base-2 logarithm, which tells us the power to which 2 must be raised to achieve \(n\).
In the given problem, the aim is to show \(a_n = \left[\log_2 n\right]\), meaning we are seeking a way the recursive sequence relates to base-2 logarithms.
Logarithmic characteristics like properties and rules bring clarity to these relationships.
  • Logarithms condense large problems into manageable ones.
  • The integral part \(\left[\log_2 n\right]\) reflects how many times \(n\) can be divided by 2 until reaching below 1.
Recognizing how recursive sequences can be represented through logarithmic functions illuminates the solution and emphasizes the efficiency of logarithms in mathematics.

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Most popular questions from this chapter

a) How many subsets \(A=\\{a, b, c, d\\} \subseteq \mathbf{Z}^{+}\), where \(a, b, c, d>1\), satisfy the property \(a \cdot b \cdot c \cdot d=\) \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19\) ? b) How many subsets \(A=\left\\{a_{1}, a_{2}, \ldots, a_{m}\right\\} \subseteq \mathbf{Z}^{+}\), where \(a_{i}>1,1 \leq i \leq m\), satisfy the property \(\prod_{i=1}^{m} a_{l}=\prod_{j=1}^{n} p_{j}\), where the \(p_{j}, 1 \leq j \leq n\), are distinct primes and \(n \geq m\) ?

a) Write a computer program (or develop an algorithm) to determine the minimum and maximum values in an array \(A[1], A[2], A[3], \ldots, A[n]\) of integers. (Here \(n \in \mathbf{Z}^{+}\) with \(n \geq 2\), and the entries in the array need not be distinct.) b) Determine the worst-case complexity function for the implementation developed in part (a).

Given 8 Pascal books, 17 FORTRAN books, 6 APL books, 12 COBOL books, and 20 BASIC books, how many of these books must we select to insure that we have 10 books dealing with the same computer language?

Give an example of a function \(f: \mathbf{Z}^{+} \rightarrow \mathbf{R}\) where \(f \in O(1)\) and \(f\) is one-to-one. (Hence \(f\) is not constant.)

Let \(a, b, c \in \mathbf{R}\) with \(a^{2}+b c \neq 0\). If \(f(x)=(a x+b)\) ? \((c x-a)\), verify that \((f \circ f)(x)=x\) for all \(x \in \mathbf{R}, x \neq a / c\).
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