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A person starts walking from home (at \(x=0\) ) toward a friend's house (at \(x=1\) ). Three-fourths of the way there, he changes his mind and starts walking back home. Three-fourths of the way home, he changes his mind again and starts walking back to his friend's house. If he continues this pattern of indecision, always turning around at the three-fourths mark, what will be the eventual outcome? A similar problem appeared in a national magazine and created a minor controversy due to the ambiguous wording of the problem. It is clear that the first turnaround is at \(x=\frac{3}{4}\) and the second turnaround is at \(\frac{3}{4}-\frac{3}{4}\left(\frac{3}{4}\right)=\frac{3}{16}\) But is the third turnaround three-fourths of the way to \(x=1\) or \(x=\frac{3}{4} ?\) The magazine writer assumed the latter. Show that with this assumption, the person's location forms a geometric series. Find the sum of the series to find where the person ends up.

Step by step solution

01

Identify the initial conditions

The person begins walking from his home at x=0 towards his friend's house at x=1. He changes direction for the first time when he is 3/4 of the way, which corresponds to the location \(x = \frac{3}{4}\). Therefore, the first term of the geometric series is \(a = \frac{3}{4}\).

02

Identify the common ratio of the geometric series

Every time the person changes his direction, he moves 3/4 of the 'remaining' distance before changing direction again. Therefore, the common ratio \(r\) of the geometric series is \(r = \frac{3}{4}\).

03

Find the Sum of the Geometric Series

Knowing the first term \(a\) and the common ratio \(r\), the sum \(S\) of an infinite geometric series can be calculated using the formula: \[S = \frac{a}{1-r}\]. Here, \(S = \frac{\frac{3}{4}}{1 - \frac{3}{4}}\) which simplifies to \(S = 3\).

04

Interpret the Result

The sum of 3 tells us that the person eventually ends up 3 units of distance from the starting point. Considering the start at 0 and the friend's house at 1, it means that the person actually goes past his friend's house by 2 units of distance and never reaches home again.

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Key Concepts

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

Infinite Series

An infinite series is a sum of an infinite sequence of terms. In the context of this problem, the person keeps moving back and forth along the path to the friend's house, forming an ongoing geometric sequence.
The sequence is considered infinite because the number of terms grows without end. Each turnaround is part of the pattern: moving 3/4 of the remaining distance in the opposite direction.
This transform into an infinite series when added up. It's crucial in understanding that the person doesn't need infinite time but rather infinite terms are considered to predict the outcome.

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to get the next term. Here, the person moves three-fourths of the distance each time they turn around.
This consistent fraction, 3/4, is the common ratio \( r \). It's key to how the distances decrease over time and is represented by the formula:

• Each subsequent movement = \( ext{Previous Movement} \times \frac{3}{4} \)

The common ratio determines the series' convergence and affects the sum of the series significantly.

Sum of Series

Calculating the sum of an infinite geometric series involves finding where the person's movement eventually leads. With geometric series, the sum \( S \) can be calculated using:
\[S = \frac{a}{1 - r}\]
where \( a \) is the first term and \( r \) is the common ratio. In this problem:

• \( a = \frac{3}{4} \)
• \( r = \frac{3}{4} \)

Substituting in:
\[S = \frac{\frac{3}{4}}{1 - \frac{3}{4}} = 3\]
This sum tells us the position at which the movements "add up" in an infinite sense, providing a final destination point.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number. In this scenario, the sequence is purely geometric:

• Starts at \( \frac{3}{4} \)
• Then \( \frac{3}{16} \)
• And so on...

The factor of 3/4 guides each new position. This is not only about calculating how far the person walks but also about how the terms form a predictable pattern that eventually leads to a determinate sum.