Question

Evelyn has a perfect balancing scale, an unlimited number of \(1-\mathrm{kg}\) weights, and one each of \(1 / 2-\mathrm{kg}, 1 / 4-\mathrm{kg}, 1 / 8-\mathrm{kg}\), and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?

Short answer

Yes, Evelyn can weigh the meteorite precisely using the infinite series of fractional weights for fine adjustments.

Step-by-step solution

Understanding the Problem

Evelyn wants to weigh a meteorite using a perfect balancing scale, an unlimited number of 1 kg weights, and a sequence of fractional weights (\(\frac{1}{2}\) kg, \(\frac{1}{4}\) kg, \(\frac{1}{8}\) kg, and so on). Our task is to determine if she can weigh the meteorite precisely.

Identifying the Weight Options

Evelyn has weights of specific values: unlimited 1 kg weights and a sequential series of weights decreasing by a factor of \(\frac{1}{2}\), forming a geometric series: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), etc.

Applying Infinite Series

The fractional weights form an infinite geometric series: \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\). The sum of an infinite geometric series \(a + ar + ar^2 + \ldots\) where \(|r| < 1\) is \(\frac{a}{1-r}\). Here, \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\), thus the sum is \(\frac{\frac{1}{2}}{1-\frac{1}{2}} = 1\).

Achieving Arbitrary Precision

Using an unlimited number of 1 kg weights allows setting a base integer weight on one side of the scale. The infinite series of fractional weights (up to a sum of 1 kg) enables the adjustment of the balance with arbitrary precision, as this series can represent any fraction between 0 and 1.

Conclusion

Evelyn can precisely measure the meteorite's weight to any desired accuracy by combining integer weights with fine adjustments from the infinite series. The relation to infinite series lies in using a convergent series to balance the scale accurately.

Key concepts

Infinite Series

An infinite series is a sequence of numbers that continues indefinitely by adding more terms. In this exercise, the sequence of fractional weights Evelyn uses forms an infinite geometric series. The series begins with a fraction, like \( \frac{1}{2} \), and continues by halving each subsequent term, forming \( \frac{1}{4} \), \( \frac{1}{8} \), and so on.
When dealing with infinite series, particularly geometric series, there is a formula to find the sum:

• If the first term is \( a \) and the common ratio is \( r \) (where \( |r| < 1 \)), the sum of the series is \( \frac{a}{1-r} \).

In Evelyn's case, this series converges to 1 when \( a = \frac{1}{2} \) and \( r = \frac{1}{2}\).
This convergence allows the fractional weights to effectively 'fill in' all possible weight increments between whole numbers.
Thus, by using this infinite series, Evelyn can measure weights with very fine precision, balancing it as needed.

Balancing Scale

A balancing scale is an old yet effective tool for measuring weight. It operates on the principle of even weight distribution. Evelyn’s scale is considered perfect, meaning it can detect the slightest of imbalances.
With a balancing scale, you can measure unknown weights against known weights by placing them on opposite sides until equilibrium is achieved. The key is the ability to balance the forces - in this case, weights - without tipping to one side.
Evelyn's challenge of weighing a meteorite involves using 1 kg weights and fine-tuning with fractional weights from her infinite series. By gradually adding smaller fractional weights, she can achieve balance to a very detailed degree. This method showcases the scale's ability to provide precision that is closely linked to understanding geometric series properties.

Precision Measurement

Precision Measurement is the practice of obtaining extremely accurate and specific measurements. In Evelyn's case, her goal is to weigh a meteorite with a high degree of precision. To achieve this, she uses not only whole number weights but also a series of fractional weights, which adds a finer level of detail.
Each fractional weight allows Evelyn to adjust by smaller and smaller amounts, providing a closer approximation of the meteorite’s true weight. By leveraging the infinite series, she enhances her ability to measure to the smallest margin of error possible.
The convergence of the geometric series to one kilogram is crucial because it means that all weight increments between integers are representable, enabling exact measures even in complex scenarios involving fractional values.

Fractional Weights

Fractional Weights are the key to achieving the precision that Evelyn desires on her balancing scale. These weights are fractions like \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), and so forth. Each subsequent weight is half the previous one, forming a sequence that helps fill the gaps between whole numbers.
In practical terms, using fractional weights means that Evelyn can make incremental adjustments to her measurements. This is particularly advantageous for achieving precision because each fraction of a kilogram is smaller than the one before, allowing for more exact balancing.
Utilizing fractional weights along with 1 kg weights, Evelyn can measure any weight with exceptional accuracy by adjusting the scale in slightly more fine-tuned steps. This capability is a direct application of the concept of geometric series, ensuring every potential weight increment is covered.