# 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
3 years, 11 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 $$Lim_{n \rightarrow \infty}$$

- 3 years, 11 months ago

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

- 3 years, 11 months ago

Thanks edited

- 3 years, 11 months ago