C Value is 00. The first sequenceI am stuck. Please help **Understanding Fibonacci Sequences and Variations**Let $ F $ be the Fibonacci sequence, which is defined as:\[F := \left( n \mapsto \begin{cases} 0, & \text{if } n = 0 \\ 1, & \text{if } n = 1 \\ F_{n-1} + F_{n-2}, & \text{if } n > 1 \end{cases} \right) : \mathbb{N} \rightarrow \mathbb{R}.\]Using the sequence definition for $ \beta $ indicated by your $ C $ value in the chart below, find the value of $ \beta(5) $.| **C value** | **Sequence** ||-------------|--------------|| 00 | $\beta := \left( n \mapsto \sum_{k=0}^{n} (F_{2k+1}) \right) : \mathbb{N} \rightarrow \mathbb{R} $ || 01 | $\beta := \left( n \mapsto (F_n)^2 + (F_{n+1})^2 \right) : \mathbb{N} \rightarrow \mathbb{R} $ || 10 | $\beta := \left( n \mapsto (F_{n+1})^2 - (F_n)^2 \right) : \mathbb{N} \rightarrow \mathbb{R} $ || 11 | $\beta := \left( n \mapsto F_n + F_{n+2} + F_{n+4} \right) : \mathbb{N} \rightarrow \mathbb{R} $ |To solve for $ \beta(5) $, use the sequence formula corresponding to your chosen $ C $ value from the table. Each variation provides a distinct method for calculating a sequence-related value based on the Fibonacci sequence.This table expands on how different mathematical functions can be applied to the Fibonacci sequence to generate new sequences or explore properties of numbers.

The sequence is β:=n↦∑k=0nF2k+1:ℕ→ℝ Then, on expanding β5=∑k=05F2k+1=F1+F3+F5+F7+F9+F11 We have…

Answered: Let F be the Fibonacci sequence: if n =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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C Value is 00. The first sequence I am stuck. Please help

**Understanding Fibonacci Sequences and Variations**

Let $ F $ be the Fibonacci sequence, which is defined as:

\[
F := \left( n \mapsto \begin{cases}
0, & \text{if } n = 0 \\
1, & \text{if } n = 1 \\
F_{n-1} + F_{n-2}, & \text{if } n > 1
\end{cases} \right) : \mathbb{N} \rightarrow \mathbb{R}.
\]

Using the sequence definition for $ \beta $ indicated by your $ C $ value in the chart below, find the value of $ \beta(5) $.

| **C value** | **Sequence** |
|-------------|--------------|
| 00 | $\beta := \left( n \mapsto \sum_{k=0}^{n} (F_{2k+1}) \right) : \mathbb{N} \rightarrow \mathbb{R} $ |
| 01 | $\beta := \left( n \mapsto (F_n)^2 + (F_{n+1})^2 \right) : \mathbb{N} \rightarrow \mathbb{R} $ |
| 10 | $\beta := \left( n \mapsto (F_{n+1})^2 - (F_n)^2 \right) : \mathbb{N} \rightarrow \mathbb{R} $ |
| 11 | $\beta := \left( n \mapsto F_n + F_{n+2} + F_{n+4} \right) : \mathbb{N} \rightarrow \mathbb{R} $ |

To solve for $ \beta(5) $, use the sequence formula corresponding to your chosen $ C $ value from the table. Each variation provides a distinct method for calculating a sequence-related value based on the Fibonacci sequence.

This table expands on how different mathematical functions can be applied to the Fibonacci sequence to generate new sequences or explore properties of numbers.

Expert Solution

Solution:

The sequence is $β : = (n \mapsto \sum_{k = 0}^{n} (F_{2 k + 1})) : ℕ \to ℝ$

Then, on expanding

$\begin{array}{rcl} β (5) & = & \sum_{k = 0}^{5} (F_{2 k + 1}) \\ = & F_{1} + F_{3} + F_{5} + F_{7} + F_{9} + F_{11} \end{array}$

We have

$\begin{array}{rcl} F_{1} & = & 1; F_{3} = 1 + 1 = 2; F_{5} = 2 + 3 = 5 \\ F_{7} & = & 5 + 8 = 13; F_{9} = 13 + 21 = 34; F_{11} = 34 + 55 = 89 \end{array}$

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