Problem 45 - Algebra 2 and Trigonometry

Question

One of the most famous sequences is the Fibonacci sequence. In this sequence, \(a_{1}=1, a_{2}=1,\) and for \(n>2, a_{n}=a_{n-2}+a_{n-1} .\) Write the first ten terms of this sequence.

Step-by-Step Solution

Answer

The first ten terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Step 1: Understand the Fibonacci Sequence Formula

The Fibonacci sequence is defined with the initial terms \(a_{1} = 1\) and \(a_{2} = 1\). For any term \(a_{n}\) with \(n > 2\), each term in the sequence is the sum of the two preceding terms: \(a_{n} = a_{n-2} + a_{n-1}\). This formula will be used to calculate the subsequent terms in the sequence.

Step 2: Calculate the 3rd Term

We need to find \(a_{3}\). According to the formula, \(a_{3} = a_{1} + a_{2} = 1 + 1 = 2\).

Step 3: Calculate the 4th Term

To find \(a_{4}\), use the formula: \(a_{4} = a_{2} + a_{3} = 1 + 2 = 3\).

Step 4: Calculate the 5th Term

Now calculate \(a_{5}\) using earlier terms: \(a_{5} = a_{3} + a_{4} = 2 + 3 = 5\).

Step 5: Calculate the 6th Term

Next, find \(a_{6}\) by summing the previous two terms: \(a_{6} = a_{4} + a_{5} = 3 + 5 = 8\).

Step 6: Calculate the 7th Term

To calculate \(a_{7}\), continue with the formula: \(a_{7} = a_{5} + a_{6} = 5 + 8 = 13\).

Step 7: Calculate the 8th Term

Compute \(a_{8}\) by adding the two preceding terms: \(a_{8} = a_{6} + a_{7} = 8 + 13 = 21\).

Step 8: Calculate the 9th Term

For \(a_{9}\), use the formula again: \(a_{9} = a_{7} + a_{8} = 13 + 21 = 34\).

Step 9: Calculate the 10th Term

Finally, find \(a_{10}\) with the formula: \(a_{10} = a_{8} + a_{9} = 21 + 34 = 55\).

Step 10: List the First Ten Terms

With the calculated terms, list the first ten terms of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Key Concepts

Recursive Sequence

A recursive sequence is a type of mathematical sequence where each term is defined as a function of one or more of its preceding terms. In simpler terms, it's like building a tower where you add one block at a time based on where the previous block is placed.
In the Fibonacci sequence, every term beyond the second one is the sum of the two terms that came before it. Here's the pattern:

• Start with known initial terms: usually \(a_1 = 1\) and \(a_2 = 1\).
• For any term \(a_n\) with \(n > 2\), define it using: \(a_n = a_{n-2} + a_{n-1}\).

This formula ensures that each term pulls from the most recent data, which is a hallmark of recursive sequences. With recursion, the future of the sequence is molded by its past, providing a connected pattern that continues indefinitely.

Arithmetic Understanding

Understanding arithmetic involves comprehending not just numbers and calculations, but also recognizing patterns and relationships between those numbers. One of the fundamental concepts in arithmetic is addition, which plays a central role in sequences like the Fibonacci sequence.
In Fibonacci, the arithmetic operation at play is simple addition. Here is what's happening:

• Each pair of terms is summed to get the next term: E.g., 1 + 1 = 2, 1 + 2 = 3, and so on.
• Observing how each number exactly results from the addition of its predecessors gives you insight into this sustained series of operations performed again and again.

Ultimately, this pattern of addition reveals a neat expression of growth and form that appears in many places in nature. As you study these concepts, you'll see that arithmetic understanding involves seeing beyond numbers to the structure they create together.

Algebraic Expression

Algebraic expressions are vital tools for representing and solving mathematical problems using variables and operations. The generic form of an algebraic expression is made up of constants, variables, and operators. In the context of sequences like Fibonacci, algebraic expressions help us understand and predict the behavior of sequences.
In specific:

• We use variables like \(a_n\) to represent any arbitrary term in the sequence, enabling a generalized form.
• The expression \(a_n = a_{n-1} + a_{n-2}\) is an algebraic tool capturing not just a single calculation but the entirety of the sequence's logic.

Algebra bridges arithmetic computations and abstract mathematical reasoning. It empowers us to articulate the Fibonacci sequence formula in a way that's universally understood, regardless of which term is being computed. Algebraic expressions serve as fundamental components in higher mathematical understanding, extending well beyond numeric addition.