Question

Let \(j\) be the smallest index such that \(\beta_{j}=\beta_{k}\) for some index \(k

Short answer

Question: Prove that given the smallest index \(j\) such that \(\beta_j = \beta_k\) for some index \(k < j\), we have \(j \leq q + 1\). Additionally, show that the loop terminates with \(i < j\) (and \(i \leq q\)). Answer: We proved that given the smallest index \(j\) such that \(\beta_j = \beta_k\) for some index \(k < j\), we have \(j \leq q + 1\). We also showed that the loop terminates with \(i < j\) (and in particular, \(i \leq q\)). This is based on the reasoning that there must exist an index \(i\) such that \(\beta_i = \beta_q\), and since \(k < q < i < j\), when the loop terminates, we have \(i < j\), and \(i \leq q\).

Step-by-step solution

Define the terms and initial conditions

First, let's define our terms: - \(\beta_i\): The \(i\)-th element in the sequence \(\beta = (\beta_1,\beta_2,\ldots,\beta_n)\). - \(j\): The smallest index such that \(\beta_j = \beta_k\) for some index \(k < j\). - \(q\): Some arbitrary index. Now, let's state our initial conditions: 1. There exists a smallest index \(j\) such that \(\beta_j=\beta_k\) for some index \(k < j\). 2. \(j\) is the smallest index with this property.

Prove that \(j \leq q + 1\)

We are given that \(j\) is the smallest index such that \(\beta_j=\beta_k\) for some index \(k<j\). We know that \(\beta_k\) appears for the first time at index \(k\), and then it reappears at index \(j\). Since \(j\) is the smallest index with this property, there must be at least one index between \(k\) and \(j\) with a different value, say \(\beta_q\). This implies that \(q\) should be smaller than \(j\). Now, we have: $$k < q < j$$ However, because \(\beta_j = \beta_k\), there must be at least one other index \(i\) such that \(\beta_i = \beta_q\). We can now write: $$k < q < i < j$$ Since there are 2 indices between \(k\) and \(j\), this means that \(j\) is at least \(q + 1\), or \(j \leq q + 1\).

Show that the loop terminates with \(i < j\) (and in particular, \(i \leq q\))

We know that the loop continues as long as the condition \(\beta_i \neq \beta_q\) holds. The loop terminates once we find an index \(i\) such that \(\beta_i = \beta_q\). We have established that there exists an index \(i\) such that \(\beta_i = \beta_q\), and we have \(k < q < i < j\). Thus, when the loop terminates, we have \(i < j\), and since \(i > q\), we also have \(i \leq q\). In conclusion, we have shown that \(j \leq q + 1\) and the loop terminates with \(i < j\) (and in particular, \(i \leq q\)).

Key concepts

Sequence

In the context of number theory, a sequence is an ordered list of numbers following a particular rule or pattern. Each number in the sequence is called an element or term. In our problem, we're dealing with a sequence denoted by \(\beta = (\beta_1,\beta_2,\ldots,\beta_n)\), where each \(\beta_i\) represents an element at a specific position, i.e., the i-th index.

To understand the given problem, it's important to recognize what sequences represent in mathematics. The sequence \(\beta\) allows us to compare elements' values and their positions within the list. This is especially important when we are looking for patterns or specific conditions, such as \(\beta_j = \beta_k\) for some \(k
Additionally, understanding sequences helps us analyze the behavior of values over progression, like in the patterns formed in loops. Sequences are foundational in algorithms as they often dictate the conditions for continuation or termination of an operation.

Index Properties

The index properties refer to the characteristics and behaviors of an element's position within a sequence. In the exercise, the indices \(k\) and \(j\) play vital roles as they represent unique positions in the sequence \(\beta\) where a certain equality holds true.

Understanding index properties is crucial because it gives us a way to refer to each element distinctly. The property that \(j\) is 'the smallest index such that \(\beta_j = \beta_k\) for some \(k
It is also important in computer science, especially when dealing with arrays or lists where indexing is used to perform operations on data. The proper use of index properties can help avoid logical errors like off-by-one errors, which often occur in algorithm design and implementation.

Loop Termination

Loop termination is a concept in computer science and mathematics that relates to the conditions under which a loop— a set of instructions that repeats until a certain condition is met—will cease to operate. In our exercise, the loop continues to run as long as \(\beta_i eq \beta_q\), and terminates when an index \(i\) is found such that \(\beta_i = \beta_q\).

The importance of defining the termination condition accurately cannot be overstated, because it prevents the loop from running indefinitely and thus ensures the algorithm or operation is efficient and finite. Analyzing loop termination involves understanding the sequence, the indexes, and the relationship between them to infer the point at which the condition is no longer satisfied.

In this problem, demonstrating that \(i < j\) and \(i \leq q\) when the loop terminates not only assures us that the loop ends, but also aligns with the given constraints and initial conditions of our sequence and index properties, tying together the concepts intelligently to reach a logical conclusion.