Question

The Fibonacci sequence starts with \(1,1,2,3,5, \ldots\) and each term is the sum of the previous two terms, \(F_{n}=F_{n-1}+F_{n-2}\) (a) Find \(F_{6}, F_{7}, F_{8}\) and \(F_{14}\). (b) Check the identity: \(F_{2 n}=F_{n}\left(F_{n+1}+F_{n-1}\right)\) for \(n=7\)

Short answer

Question: Calculate the Fibonacci numbers \(F_6, F_7, F_8\), and \(F_{14}\) and check the validity of the identity \(F_{2n} = F_n(F_{n+1}+F_{n-1})\) for \(n = 7\). Answer: The calculated Fibonacci numbers are \(F_6 = 8\), \(F_7 = 13\), \(F_8 = 21\), and \(F_{14} = 377\). The given identity holds true for \(n = 7\), as we found that \(F_{14} = F_7(F_8 + F_6) = 377\).

Step-by-step solution

(Step 1: Calculating Fibonacci numbers)

Given the Fibonacci sequence and the rule that defines the terms in the sequence \(F_{n}=F_{n-1}+F_{n-2}\) We are asked to find \(F_6 , F_7 , F_8\) and \(F_{14}\) where \(F_1 = F_2 = 1\) To find them, we can simply follow the rule and add the two previous numbers to get the next one: \(F_3 = F_2 + F_1 = 1 + 1 = 2\) \(F_4 = F_3 + F_2 = 2 + 1 = 3\) \(F_5 = F_4 + F_3 = 3 + 2 = 5\) Now we can find the requested Fibonacci numbers: \(F_6 = F_5 + F_4 = 5 + 3 = 8\) \(F_7 = F_6 + F_5 = 8 + 5 = 13\) \(F_8 = F_7 + F_6 = 13 + 8 = 21\) To find \(F_{14}\), we need to continue the sequence \(F_9 = F_8 + F_7 = 21 + 13 = 34\) \(F_{10} = F_9 + F_8 = 34 + 21 = 55\) \(F_{11} = F_{10} + F_9 = 55 + 34 = 89\) \(F_{12} = F_{11} + F_{10} = 89 + 55 = 144\) \(F_{13} = F_{12}+ F_{11} = 144 + 89 = 233\) \(F_{14} = F_{13} + F_{12} = 233 + 144 = 377\) So we have: \(F_6 = 8\), \(F_7 = 13\), \(F_8 = 21\), and \(F_{14} = 377\)

(Step 2: Checking the identity)

We are given the identity: \(F_{2 n}=F_{n}\left(F_{n+1}+F_{n-1}\right)\) We need to check it for \(n=7\). First, we need to find \(F_{2\cdot7}\), which we have already calculated as \(F_{14} = 377\). Now, we need to check the right-hand side of the identity: \(F_{n}\left(F_{n+1}+F_{n-1}\right)\) For \(n=7\), we have already found \(F_7 = 13\), \(F_6 = 8\), and \(F_8 = 21\). Now substituting these values into the identity: \(F_{7}\left(F_{8}+F_{6}\right) = 13 \cdot (21 + 8)\) Calculating this expression: \(13 \cdot (21 + 8) = 13 \cdot 29 = 377\) Since we have shown that \(F_{14} = 377\) and \(F_{7}\left(F_{8}+F_{6}\right) = 377\), the given identity holds true for \(n=7\).

Key concepts

recursive sequences

A recursive sequence is a sequence of numbers where each term is derived based on one or more of the preceding terms through a specific rule or formula. In the case of the Fibonacci sequence, this rule is expressed as \(F_{n} = F_{n-1} + F_{n-2}\). This means that each term is the sum of the two terms that come before it in the sequence. Such sequences are valuable because they allow us to generate complex and large patterns from simple recursive relations using relatively straightforward computations.
Recursive sequences like the Fibonacci sequence often arise in various fields such as computer science, biology, and mathematics. They provide insights into natural phenomena and can be used to model growth patterns, populations, and even financial markets. The simplicity of their construction is what makes them powerful in both theoretical understanding and practical applications.
When working with recursive sequences, it's crucial to understand the initial values, as they determine the entire sequence. In the Fibonacci sequence, the initial terms are both 1, setting the foundation for all subsequent numbers.

mathematical identities

Mathematical identities are equations that hold true for all values involved, providing valuable shortcuts and checks in algebra and calculus. In our exercise, the identity \(F_{2 n} = F_{n}(F_{n+1} + F_{n-1})\) is used to verify computations related to the Fibonacci sequence.
Such identities allow mathematicians to relate different aspects of complex problems, proving or simplifying them significantly. They often help in understanding deeper properties of mathematical structures or patterns, such as those seen in number theory – where Fibonacci identities are frequently used.
By proving the identity in the exercise, we see how mathematics uses logic and known properties to verify seemingly independent calculations, ensuring both accuracy and understanding. This is especially beneficial in ensuring that our solutions conform to the inherent mathematical structures when working with sequences and series.

sequence terms

In any mathematical sequence, terms are the individual numbers that make up the sequence. For the Fibonacci sequence, these terms begin with 1, 1, and continue as 2, 3, 5, and so on, where each term is a specific value appearing in its respective order within the sequence.
Understanding sequence terms involves recognizing their placement as well as their value, which helps in identifying patterns and relationships among terms. This is crucial in solving problems related to sequences, such as determining unknown terms and verifying mathematical identities related to sequences.
Learning to find Fibonacci sequence terms using its recursive rule involves simple arithmetic: adding the two previous terms to get the next. As illustrated in the exercise, starting with known values and using the rule to advance within the sequence showcases how easily sequence terms can be generated manually or computationally.

algebraic problem solving

Algebra is a powerful tool in solving wide-ranging mathematical problems, including those involving sequences. Algebraic problem solving is about manipulating expressions and equations to find unknown values, check identities, or derive general formulas. In dealing with sequences like the Fibonacci series, algebra assists in verifying identities or conjectures—and more importantly, in deriving terms from given recursive formulas.
By utilizing algebraic methods, we can approach these problems step-by-step, systematically calculating terms and proving identities. For instance, substituting known values into an identity and simplifying, as done in the problem's step-by-step solution, allows for a structured approach to confirming results.
Through algebraic problem solving, mathematical explorations become more than just arithmetic tasks; they evolve into methods of inferring patterns, relationships, and deeper truths about numbers. This exemplifies the interactive nature of algebra with various mathematical realms, offering a robust foundation for tackling sequence problems.