Problem 77 - Contemporary Precalculus

Question

Deal with the Fibonacci sequence \(\left\\{a_{n}\right\\}\) that was discussed in Example 6. (a) List the first 10 terms of the Fibonacci sequence. (b) List the first 10 partial sums of the sequence. (c) Do the partial sums follow an identifiable pattern?

Step-by-Step Solution

Answer

Question: Analyze and provide a pattern based on the first 10 partial sums of the Fibonacci sequence. Answer: Analyzing the first 10 partial sums, we find that the difference between two consecutive partial sums equals the corresponding Fibonacci term. In other words, \(S_{n} - S_{n-1} = a_n\).

Step 1: Find the first 10 Fibonacci terms

The Fibonacci sequence is defined as follows: \(a_1 = 1,\, a_2 = 1,\, a_n = a_{n-1} + a_{n-2}\), for \(n > 2\). Using this definition, let's find the first 10 terms: \(a_1 = 1\) \(a_2 = 1\) \(a_3 = a_1 + a_2 = 1 + 1 = 2\) \(a_4 = a_2 + a_3 = 1 + 2 = 3\) \(a_5 = a_3 + a_4 = 2 + 3 = 5\) \(a_6 = a_4 + a_5 = 3 + 5 = 8\) \(a_7 = a_5 + a_6 = 5 + 8 = 13\) \(a_8 = a_6 + a_7 = 8 + 13 = 21\) \(a_9 = a_7 + a_8 = 13 + 21 = 34\) \(a_{10} = a_8 + a_9 = 21 + 34 = 55\) Thus, the first 10 terms are: \(1, 1, 2, 3, 5, 8, 13, 21, 34, 55\).

Step 2: Calculate the first 10 partial sums of the Fibonacci sequence

A partial sum is the sum of the first \(n\) terms in a sequence. We found the first 10 Fibonacci terms in step 1, so now let's calculate the first 10 partial sums, denoted as \(S_n\): \(S_1 = a_1 = 1\) \(S_2 = a_1 + a_2 = 1 + 1 = 2\) \(S_3 = a_1 + a_2 + a_3 = 1 + 1 + 2 = 4\) \(S_4 = a_1 + a_2 + a_3 + a_4 = 1 + 1 + 2 + 3 = 7\) \(S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 1 + 1 + 2 + 3 + 5 = 12\) \(S_6 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 1 + 1 + 2 + 3 + 5 + 8 = 20\) \(S_7 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 = 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33\) \(S_8 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 54\) \(S_9 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88\) \(S_{10} = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 + a_{10} = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143\)

Step 3: Analyzing the pattern in partial sums

Now that we have calculated the first 10 partial sums, let's analyze their pattern: \(S_1 = 1\) \(S_2 = 2\) \(S_3 = 4\) \(S_4 = 7\) \(S_5 = 12\) \(S_6 = 20\) \(S_7 = 33\) \(S_8 = 54\) \(S_9 = 88\) \(S_{10} = 143\) We can see that the difference between two adjacent sums seems to be related to the Fibonacci sequence itself. Let's look into this pattern more deeply: - \(S_2 - S_1 = a_2 = 1\) - \(S_3 - S_2 = a_3 = 2\) - \(S_4 - S_3 = a_4 = 3\) - \(S_5 - S_4 = a_5 = 5\) - \(S_6 - S_5 = a_6 = 8\) So we can see a pattern emerging: the difference between two consecutive partial sums is equal to the corresponding Fibonacci term, i.e., \(S_{n} - S_{n-1} = a_n\).

Key Concepts

Precalculus and Mathematical Sequences

Precalculus lays the groundwork for understanding sequences and series, which are fundamental concepts in higher-level mathematics. A sequence is an ordered list of numbers, where each number is called a term. In precalculus, students learn to identify the type of sequence based on its rule for term generation. Common sequences include arithmetic and geometric sequences, but another captivating type of sequence is the Fibonacci sequence.

This sequence exhibits a recursive pattern where each term after the initial two is the sum of the two preceding terms. The Fibonacci sequence is a perfect example of how simple rules can create complex and beautiful mathematical patterns, offering a practical application for precalculus students to recognize patterns and develop their problem-solving skills.

Understanding Fibonacci Terms

The Fibonacci sequence, discovered by Leonardo of Pisa, starts with two ones, and each subsequent term is the sum of the two preceding ones. Precisely, the sequence begins as follows: 1, 1, 2, 3, 5, 8, and so on. Students should note that the sequence is inherently recursive - each term is built upon the framework established by its ancestors.

Learning the Fibonacci sequence reinforces the concept of recursion and pattern recognition. These skills are essential not just in mathematics, but in various fields such as computer science, economics, and the natural sciences, where patterns emerge and can be described recursively.

Partial Sums in Sequences

Partial sums are the sum of the first 'n' terms of a sequence, vital for analyzing series. In the context of the Fibonacci sequence, each partial sum is obtained by adding up all terms up to a certain position. This concept teaches students about accumulation and growth, which are valuable for understanding more complex functions in calculus, like integrals.

For the Fibonacci series, the partial sums themselves create a new sequence that retains a connection to the original Fibonacci terms. Recognizing this relationship allows students to deepen their understanding of sequences and how disparate mathematical concepts are often linked.

Identifying Sequence Patterns

The ability to identify patterns in sequences is a skill that precalculus hones. By calculating the differences between subsequent partial sums of the Fibonacci sequence, students encounter a beautiful emergent pattern: the differences themselves align with the Fibonacci sequence.

This reveals a fascinating property: the pattern within the sequence is self-replicating even in the transformations of the sequence, such as its partial sums. Analyzing sequence patterns encourages abstract thinking and helps students make connections to broader mathematical concepts, forging an essential link between discrete mathematics and continuous phenomena.