Problem 33 - Advanced Calculus

Question

Let \(A=(0,1)\) and \(B=(1,0)\). Let \(P_{1}\) be any point in the plane, and construct a sequence \(\left\\{P_{n}\right\\}\), with \(P_{1}\) as its first term, as follows: Let \(Q_{1}=\) midpoint of \(A P_{1}\) and \(P_{2}=\) midpoint of \(B Q_{1}\) : then. let \(Q_{2}=\) midpoint of \(A P_{2}\) and \(P_{3}=\) midpoint of \(B Q_{2}\), and so on. Prove that \(\left\\{P_{n} \mid\right.\) converges.

Step-by-Step Solution

Answer

The sequence \(\{P_{n}\}\) converges as the distance between any two successive terms (i.e., \( P_{n+1} \) and \( P_{n} \)) is progressively decreasing and tending towards zero, indicating that the terms are getting closer and closer which is the hallmark of a converged sequence.

Step 1: Understanding the Sequence Construction

Let's start by constructing the sequence from the given criteria. We start with a point \( P_{1} \). The next point in the sequence, denoted as \( P_{2} \), is determined by firstly finding the midpoint \( Q_{1} \) of the line segment joining \( A = (0,1) \) and \( P_{1} \). Following that, \( P_{2} \) is obtained as the midpoint of the line segment connecting \( B = (1,0) \) and \( Q_{1} \). This process is performed iteratively to generate a series of points.

Step 2: Seeing the Pattern and Hypothesis Formulation

Having understood the criteria for creating the sequence, the next step is looking for a pattern. If you iterate this method, you might observe that the new point at each step always falls inside the triangle formed by \( AB \), and the current and preceding point. This suggests the sequence might be converging towards a certain point.

Step 3: Proving Convergence of the Sequence

Now to prove the convergence of the sequence, we need to establish that the difference between any two successive terms is progressively reducing and tending towards zero. The key here is to see that the distance to next point in the sequence is always half the distance to the preceding one, since each new point is defined as a midpoint which effectively halves the distance. Hence, we can infer that the successive terms in our sequence are getting progressively closer, thus indicating that the sequence is indeed converging.

Key Concepts

Midpoint Theorem

The midpoint theorem is a fundamental concept in geometry. It states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. In the context of the sequence convergence problem, the theorem is applied iteratively. By always finding midpoints, new points are positioned closer to the convergence point. When you find the midpoint of a line, you're effectively averaging the coordinates. This operation divides the length of the segment into two equal parts.

• For a segment with endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the midpoint \(M\) is \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).
• This results in the point being equidistant from both points A and B.
• Midpoints play a crucial role in reducing the distance between points in the sequence.

This theorem ensures that each step of the sequence smoothly transitions the points closer together, indicating the potential for convergence.

Iterative Process

The iterative process is when a procedure is repeated, each time using the result from the previous step. This forms a central part of problem-solving in mathematics. Here, an iterative process involves repeatedly finding midpoints of specific line segments. The present exercise asks us to find new points in a sequence based on prior elements.

• Start with a given point \(P_1\).
• Find the midpoint \(Q_1\) between a fixed point \(A\) and \(P_1\).
• Next, \(P_2\) is the midpoint between another fixed point \(B\) and \(Q_1\).
• Continue this approach iteratively: each point \(P_n\) relies on finding midpoints, reducing the gap between successive points.

This iterative nature is crucial because it creates a predictable and systematic approach to identifying the sequence's components. With each iteration, points are brought closer together, moving steadily towards convergence.

Geometric Sequence

A geometric sequence is a mathematical progression where each term is derived by multiplying the previous one by a fixed, non-zero number known as the common ratio. Although the given problem doesn’t strictly involve a traditional geometric sequence, it shares similar characteristics, specifically in convergence.

• The concept of halving distances in midpoints resembles the idea of a consistent ratio reduction, akin to a geometric progression.
• The sequence's convergence is due to the fact that distances exponentially decrease towards zero, like a geometric pattern.
• Here, instead of multipliers, the distance is halved, keeping the sequence bounded within a specific range.

Understanding this implies recognizing the capacity of systematically reducing distances to zero, which is crucial for guaranteeing convergence in a sequence. This aspect of diminishing separation keeps the points inevitably drawing near to a single convergence point.