# Play with Fibonacci

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

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

Note by U Z
5 years, 9 months ago

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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 $Lim_{n \rightarrow \infty}$

- 5 years, 9 months ago

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

- 5 years, 9 months ago

Thanks edited

- 5 years, 9 months ago