Play with Fibonacci

This is the continuation of the note by Fahim Shariar Shakkhor

The $\text{Fibonacci Sequence}$ is a sequence of integers where the first $\displaystyle 2$ term are $\displaystyle 0$ and $\displaystyle 1$ , and each subsequent term is the sum of the previous two numbers.

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 233 , 377, ...$

The $n$ -th Fibonacci number is denoted by $F_n$ and the sequence is defined by the recurrence relation

$\large F_n = F_{n-1} + F_{n-2}.$

Here, $F_0 = 0,\enspace F_1 = 1,\enspace F_2 = F_1 + F_0 = 1$ etc.

$\large F_n = \dfrac{ \left( \dfrac{1+\sqrt{5}}{2} \right)^{n}- \left(\dfrac{1-\sqrt{5}}{2} \right)^{n} }{\sqrt{5}}$

$\text{Problem 1:}~$ (By - Jake Lai) Prove that for all - $\large |x| < \phi^{-1}$

$\large \displaystyle \dfrac{x}{1-x-x^{2}} = \sum_{k=0}^{\infty} F_{k}x^{k}$

$\text{Problem 2:} ~~ \large \displaystyle \sum_{i=0}^{n} \dfrac{F_{i}}{i}$

$\text{Problem 3:} ~~ \large \displaystyle \sum_{i=1}^{n} \frac{1}{F_{i+1}\times F_{i}}$

$\text{Problem 4:} ~~\large \displaystyle \sum_{i=1}^{n} \frac{1}{F_{i}}$

$\text{Problem 5:}$ By Azhaghu Roopesh M

$\large \displaystyle \sum_{p=0}^{n-1} \binom{n-p}{p} = F_{n+1}$

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Math

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2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

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Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
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1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Problem 4:

Given below is a Java Code (Netbeans IDE)

Sorry, I couldn't get the code arranged properly using LaTex as @Brock Brown does, so I had to upload a pic .

@megh choksi , I don't think an accurate partial sum can be developed, so I just let the upper bound be a very large number since I guessed that it will converge(yes it does !) .

Can you write a Mathematical solution ? Take the $Lim_{n \rightarrow \infty}$

A Former Brilliant Member - 6 years, 4 months ago

@megh choksi Also remember $F_0 = 0$ so change the lower bounds of the summations wherever necessary :)

Thanks edited

U Z - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$