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F2 + F4 + F6 + F3 + · · · + F2n = F2n+1 – 1. ... -

F2 + F4 + F6 + F3 + · · · + F2n = F2n+1 – 1. ... -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Formula Explanation on Fibonacci Sequence** This formula pertains to a property of the Fibonacci sequence, where each term is the sum of the two preceding ones, typically starting with F₁ = 1 and F₂ = 1. **Formula Description:** The expression is given as follows: \( F_2 + F_4 + F_6 + F_8 + \cdots + F_{2n} = F_{2n+1} - 1 \). **Key Components:** - \( F_k \): Represents the k-th Fibonacci number. - The left side of the equation represents the sum of even-indexed Fibonacci numbers up to \( F_{2n} \). - The right side of the equation states that this sum is equal to \( F_{2n+1} - 1 \). **Understanding the Concept:** This identity highlights an interesting property of Fibonacci numbers, showing a relationship between the sum of even-indexed Fibonacci numbers and another Fibonacci number minus one. It is a useful tool in proofs and mathematical explorations involving Fibonacci sequences. **Applications:** Such identities are useful in algorithm design, computer science, and mathematical proofs where Fibonacci numbers frequently appear. --- This explanation provides a concise breakdown of the formula and its relevance to the Fibonacci sequence.
Transcribed Image Text:**Formula Explanation on Fibonacci Sequence** This formula pertains to a property of the Fibonacci sequence, where each term is the sum of the two preceding ones, typically starting with F₁ = 1 and F₂ = 1. **Formula Description:** The expression is given as follows: \( F_2 + F_4 + F_6 + F_8 + \cdots + F_{2n} = F_{2n+1} - 1 \). **Key Components:** - \( F_k \): Represents the k-th Fibonacci number. - The left side of the equation represents the sum of even-indexed Fibonacci numbers up to \( F_{2n} \). - The right side of the equation states that this sum is equal to \( F_{2n+1} - 1 \). **Understanding the Concept:** This identity highlights an interesting property of Fibonacci numbers, showing a relationship between the sum of even-indexed Fibonacci numbers and another Fibonacci number minus one. It is a useful tool in proofs and mathematical explorations involving Fibonacci sequences. **Applications:** Such identities are useful in algorithm design, computer science, and mathematical proofs where Fibonacci numbers frequently appear. --- This explanation provides a concise breakdown of the formula and its relevance to the Fibonacci sequence.
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