Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Bode's sequence Bode's sequence, defined by $$ a_{1}=0.4, \quad a_{k}=0.1\left(3 \cdot 2^{k-2}+4\right) \text { for } k \geq 2, $$ can be used to approximate distances of planets from the sun. These distances are measured in astronomical units, with \(1 \mathrm{AU}=93,000,000 \mathrm{mi}\). For example, the third term corresponds to Earth and the fifth term to the minor planet Ceres. Approximate the first five terms of the sequence.

Step-by-Step Solution

Answer

The first five terms are 0.4, 0.7, 1.0, 1.6, 2.8.

Step 1: Identify Initial Term

The first term of the sequence is given as \( a_1 = 0.4 \). This will form the base term from which subsequent terms will build.

Step 2: Determine the Formula for Subsequent Terms

For \( k \geq 2 \), the terms of the sequence are generated by the formula \( a_k = 0.1(3 \cdot 2^{k-2} + 4) \). This formula will be used to calculate the terms \( a_2 \) through \( a_5 \).

Step 3: Calculate the Second Term

Substitute \( k = 2 \) into the formula:\[ a_2 = 0.1(3 \cdot 2^{2-2} + 4) = 0.1(3 \cdot 1 + 4) = 0.1(7) = 0.7 \]

Step 4: Calculate the Third Term

Substitute \( k = 3 \) into the formula:\[ a_3 = 0.1(3 \cdot 2^{3-2} + 4) = 0.1(3 \cdot 2 + 4) = 0.1(10) = 1.0 \]

Step 5: Calculate the Fourth Term

Substitute \( k = 4 \) into the formula:\[ a_4 = 0.1(3 \cdot 2^{4-2} + 4) = 0.1(3 \cdot 4 + 4) = 0.1(16) = 1.6 \]

Step 6: Calculate the Fifth Term

Substitute \( k = 5 \) into the formula:\[ a_5 = 0.1(3 \cdot 2^{5-2} + 4) = 0.1(3 \cdot 8 + 4) = 0.1(28) = 2.8 \]

Step 7: Conclusion: Listing the First Five Terms

The first five terms of Bode's sequence are \( a_1 = 0.4 \), \( a_2 = 0.7 \), \( a_3 = 1.0 \), \( a_4 = 1.6 \), and \( a_5 = 2.8 \).

Key Concepts

Astronomical Units

Astronomical Units (AU) are a standard unit of measurement used to describe distances within our solar system. One Astronomical Unit is equivalent to the average distance between the Earth and the Sun, which is approximately 93 million miles or about 149.6 million kilometers. This measurement provides a simple way to express and compare distances between planets and other celestial objects without dealing with huge numbers. Understanding the concept of AU is crucial when learning about planetary distances and models like Bode's sequence that approximate these distances. Here are a few facts about astronomical units:

• 1 AU is the average distance from the Earth to the Sun.
• It simplifies the representation of distances within the solar system.
• It provides a bridge between smaller units like kilometers and the vast expanses of space.
• This unit is widely used in astronomy to discuss planetary orbits and celestial bodies.

The use of AU enables astronomers and students to grasp large distances more intuitively. For instance, when Bode's rule talks about Earth being 1.0 AU from the Sun, it instantly communicates that exact relationship in a manageable way.

Planetary Distances

Understanding planetary distances is essential for exploring our solar system and beyond. Distances between planets can vary based on their orbits, but Bode's sequence provides a rough estimation. In the context of Bode's sequence, each term gives an approximate distance in Astronomical Units from the Sun. This sequence presents a pattern that reflects the spacing of planetary orbits. Planetary distances are typically calculated in terms of AU to maintain clarity and consistency. With Bode's rule:

• The first term corresponds to the innermost object, Mercury.
• Following terms represent planets like Earth, Mars, and the asteroid belt (e.g., Ceres).
• This simplification helps in determining relative distances easily.
• Understanding these distances can aid in visualizing the solar system's vastness.

While Bode's sequence is not entirely accurate for all bodies, it effectively provides a framework for understanding solar scale. For example, the Earth's distance from the sun in Bode's sequence is around 1.0 AU, close to its actual average orbit.

Sequence Calculation

Sequence calculation involves deriving terms based on a mathematical formula. Bode's sequence offers an interesting method to approximate planetary distances by defining each term through calculations. In general terms:

• The sequence starts with a given first term, such as 0.4 AU for Mercury.
• Each subsequent term is calculated using a specific formula: \( a_k = 0.1(3 \cdot 2^{k-2} + 4) \).
• This formula involves exponential growth with factors of 2 and arithmetic adjustments (adding 4 then multiplying by 0.1).

To calculate any term in Bode's sequence:1. Substitute \( k \) with the desired term number in the sequence.2. Follow the operations indicated in the formula (first exponentiation, then multiplication, addition, and finally multiplication by 0.1).Let's take an example for \( k = 4 \):1. Substitute in the formula: \( a_4 = 0.1(3 \cdot 2^{4-2} + 4) \)2. Calculate exponent: \( 2^{4-2} = 4 \)3. Perform multiplication and addition: \( 3 \cdot 4 + 4 = 16 \)4. Multiply by 0.1: \( 0.1 \cdot 16 = 1.6 \)This detailed step-by-step method ensures accuracy in computing terms, allowing a deeper understanding of planetary approximations through sequences.