Problem 80 - Contemporary Precalculus

Question

Deal with the Fibonacci sequence \(\left\\{a_{n}\right\\}\) that was discussed in Example 6. Verify that \(\left(a_{n}\right)^{2}=a_{n+1} a_{n-1}+(-1)^{n-1}\) for \(n=2, \ldots, 10\).

Step-by-Step Solution

Answer

Question: Verify the formula \((a_n)^2 = a_{n+1}a_{n-1} + (-1)^{n-1}\) for \(n=2, \ldots, 10\) in the Fibonacci sequence. Answer: We have verified the formula \((a_n)^2 = a_{n+1}a_{n-1} + (-1)^{n-1}\) for \(n=2, \ldots, 10\) using the calculated values of the Fibonacci sequence. For all values of \(n\), both sides of the equation are equal, thus confirming the validity of the formula.

Step 1: Find the Fibonacci sequence values for n=1 to n=11.

To verify the formula for \(n=2, \ldots, 10\), we need to calculate the Fibonacci sequence values from \(a_1\) to \(a_{11}\). We use the recurrence relation \(a_n = a_{n-1} + a_{n-2}\) starting from \(a_1 = 1\) and \(a_2 = 1\). Here are the Fibonacci sequence values: \(a_1 = 1\) \(a_2 = 1\) \(a_3 = a_2 + a_1 = 1+1 = 2\) \(a_4 = a_3 + a_2 = 2+1 = 3\) \(a_5 = a_4 + a_3 = 3+2 = 5\) \(a_6 = a_5 + a_4 = 5+3 = 8\) \(a_7 = a_6 + a_5 = 8+5 = 13\) \(a_8 = a_7 + a_6 = 13+8 = 21\) \(a_9 = a_8 + a_7 = 21+13 = 34\) \(a_{10} = a_9 + a_8 = 34+21 = 55\) \(a_{11} = a_{10} + a_9 = 55+34 = 89\)

Step 2: Verify the formula for n=2 to n=10.

We will now verify the formula \((a_n)^2 = a_{n+1}a_{n-1} + (-1)^{n-1}\) for \(n=2, \ldots, 10\) using the calculated values of the Fibonacci sequence from step 1. For \(n=2\): \((a_2)^2 = 1^2 = 1\) \(a_{3}a_1 + (-1)^1 = 2*1-1 = 1\) Both values are equal. For \(n=3\): \((a_3)^2 = 2^2 = 4\) \(a_{4}a_2 + (-1)^2 = 3*1+1 = 4\) Both values are equal. For \(n=4\): \((a_4)^2 = 3^2 = 9\) \(a_{5}a_3 + (-1)^3 = 5*2-1 = 9\) Both values are equal. For \(n=5\): \((a_5)^2 = 5^2 = 25\) \(a_{6}a_4 + (-1)^4 = 8*3+1 = 25\) Both values are equal. For \(n=6\): \((a_6)^2 = 8^2 = 64\) \(a_{7}a_5 + (-1)^5 = 13*5-1 = 64\) Both values are equal. For \(n=7\): \((a_7)^2 = 13^2 = 169\) \(a_{8}a_6 + (-1)^6 = 21*8+1 = 169\) Both values are equal. For \(n=8\): \((a_8)^2 = 21^2 = 441\) \(a_{9}a_7 + (-1)^7 = 34*13-1 = 441\) Both values are equal. For \(n=9\): \((a_9)^2 = 34^2 = 1156\) \(a_{10}a_8 + (-1)^8 = 55*21+1 = 1156\) Both values are equal. For \(n=10\): \((a_{10})^2 = 55^2 = 3025\) \(a_{11}a_9 + (-1)^9 = 89*34-1 = 3025\) Both values are equal. Thus, we have verified the formula \((a_n)^2 = a_{n+1}a_{n-1} + (-1)^{n-1}\) for \(n=2, \ldots, 10\).

Key Concepts

Recurrence Relation

A recurrence relation is a way of defining a sequence where each term is formulated from its preceding terms. In the context of the Fibonacci sequence, it allows us to determine any term in the series based on the values of the two preceding terms. Specifically, the Fibonacci sequence is defined by the recurrence relation:

• \[ a_n = a_{n-1} + a_{n-2} \]

This formula tells us that each term (\( a_n \)) is the sum of the two preceding terms, starting from the initial values \( a_1 = 1 \) and \( a_2 = 1 \).
This concept is simple yet powerful. Recurrence relations can generate sequences systematically and are often used in computer algorithms for tasks like sorting or generating complex mathematical models.
Understanding and applying recurrence relations involves identifying initial conditions and using the relation repeatedly to generate subsequent terms.

Mathematical Proof

Mathematical proofs provide a logical demonstration that a particular statement is universally true. In mathematics, they’re essential for validating equations and theorems. For the Fibonacci formula verification, we use a proof approach to confirm:

• \[ (a_n)^2 = a_{n+1}a_{n-1} + (-1)^{n-1} \]

By checking specific values, we ensure the formula holds true for each integer from \( n=2 \) to \( n=10 \). For each index, both sides of the equation are calculated separately and then compared. For example, given \( n=2 \), we found that both sides equaled 1.
This kind of proof is often called a 'verification by cases', where the statement is shown to be true in each individual scenario specified by the problem. Mathematical proofs reassure us that our mathematical conclusions are accurate by relying on foundational mathematical truths.

Sequence Verification

Sequence verification involves checking that each part of a sequence fits a particular formula or set of rules. For the Fibonacci sequence, verification ensures that the derived formula aligns accurately with expected sequence values.
By calculating each term's square (\( (a_n)^2 \)) and comparing it to the right-hand side of the formula (\( a_{n+1}a_{n-1} + (-1)^{n-1} \)), we confirm validity. Verification is not just a mechanical process but a thoughtful one — it involves understanding both components of the equation and how sequence values interact. This methodical approach equips students with tools for tackling more complex mathematical relations in future studies.
The concept of sequence verification extends beyond Fibonacci numbers, applying to any sequences where consistency with a given rule must be checked.

Precalculus

Precalculus is the foundation for higher-level math concepts and typically involves the study of functions and limits. The exploration of sequences like the Fibonacci sequence often forms part of precalculus studies.
Understanding how sequences are formed, verified, and proven can offer deep insights into mathematical reasoning. Precalculus exposes students to various mathematical patterns and prepares them for calculus by developing problem-solving skills through analysis and function definition. In problems like the Fibonacci sequence, students learn to create formulas, engage with proofs, and verify sequences — all critical skills in advanced mathematics.
Incorporating real examples of recurrence relations and verification, such as the Fibonacci sequence task, helps students appreciate the relevance of precalculus outside pure mathematics, often appearing in fields such as computer science, physics, and even biology.