Problem 92 - Algebra and Trigonometry

Question

Droste Effect The Droste Effect, named after the image on boxes of Droste cocoa powder, refers to an image that contains within it a smaller version of the image, which in turn contains an even smaller version, and so on. If each version of the image is \(\frac{1}{5}\) the height of the previous version, the height of the \(n\) th version is given by \(a_{n}=\frac{1}{5} a_{n-1}\). Suppose a Droste image on a package has a height of 4 inches. How tall would the image be in the 6 th version?

Step-by-Step Solution

Answer

0.00128 inches

Step 1 - Understand the Problem

The Droste Effect creates a repeating image where each version of the image is \(\frac{1}{5}\) the height of the previous one. We need to determine the height of the 6th version if the orginal height is 4 inches.

Step 2 - Identify the Formula

The height of the nth version is given recursively by \(a_{n} = \frac{1}{5}a_{n-1}\). We know the first image \(a_{1}\) has a height of 4 inches.

Step 3 - Calculate the 2nd Version Height

Using the formula, \(a_{2} = \frac{1}{5}a_{1}\), \ a_{2} = \frac{1}{5}(4) = 0.8 \ inches.

Step 4 - Calculate the 3rd Version Height

Using the formula, \(a_{3} = \frac{1}{5}a_{2}\), \ a_{3} = \frac{1}{5}(0.8) = 0.16 \ inches.

Step 5 - Calculate the 4th Version Height

Using the formula, \(a_{4} = \frac{1}{5}a_{3}\), \ a_{4} = \frac{1}{5}(0.16) = 0.032 \ inches.

Step 6 - Calculate the 5th Version Height

Using the formula, \(a_{5} = \frac{1}{5}a_{4}\), \ a_{5} = \frac{1}{5}(0.032) = 0.0064 \ inches.

Step 7 - Calculate the 6th Version Height

Using the formula, \(a_{6} = \frac{1}{5}a_{5}\), \ a_{6} = \frac{1}{5}(0.0064) = 0.00128\ inches.

Key Concepts

recursive sequence

A recursive sequence is a sequence of numbers where each term is defined as a function of its preceding terms. In the context of the Droste Effect, the height of each version of the image forms a recursive sequence. We can write the formula as: \[ a_{n} = \frac{1}{5}a_{n-1} \]Here, each image is one-fifth the height of the previous image. The first term, or the height of the original image, is given as 4 inches. This original height is denoted by \( a_{1} \). To find the height of the nth version, we repeatedly apply the formula.

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 constant called the common ratio. In the Droste Effect exercise, the heights of the images form a geometric progression with the common ratio of \( \frac{1}{5} \). The formula to find any term in a geometric progression when the first term \(a_{1}\) and the common ratio \(r\) are known is: \[a_{n} = a_{1} \cdot r^{(n-1)} \]Applying this to our problem, we get: \[a_{6} = 4 \cdot (\frac{1}{5})^{5} \]This provides an alternative way to find the height of the sixth version directly.

repeating images

The Droste Effect involves repeating images within themselves. Each smaller image is a reduction of the original image, maintaining the same proportions. This concept is crucial for understanding how the heights change with each iteration. If the original image height is given, each smaller version height can be calculated by repeatedly applying the reduction factor of \(\frac{1}{5}\). As you continue to reduce the size, the images become smaller and smaller but never entirely disappear. This effect visually demonstrates an infinite loop in a finite manner.

height calculation

Calculating the height of each version involves applying recursive or geometric formulas step by step. Starting from the initial height of 4 inches:

• Second version height: \(4 \times \frac{1}{5} = 0.8\) inches
• Third version height: \(0.8 \times \frac{1}{5} = 0.16\) inches
• Fourth version height: \(0.16 \times \frac{1}{5} = 0.032\) inches
• Fifth version height: \(0.032 \times \frac{1}{5} = 0.0064\) inches
• Sixth version height: \(0.0064 \times \frac{1}{5} = 0.00128\) inches

This iterative process of height calculation exemplifies the Droste Effect mathematically.