Question

What is the name of these sequences?

0,1,1,2,3,5,8,13,21,34…………

Answer 1

$\large\text{\underline{What is this sequence?}}$

The name is Fibonacci sequence.

$\large\text{\underline{Explanation of the sequence?}}$

$F_{n}$(term $n$ of the Fibonacci sequence)

In this question, the two initial values are taken as $F_{0}=1$ and $F_{1}=1$, or sometimes as $F_{1}=F_{2}=1$.

The recurrence relation is $F_{n+2}=F_{n}+F_{n+1}$.

The ratio of two consecutive terms $\dfrac{F_{n+1}}{F_{n}}$ converges to $\dfrac{1+\sqrt{5}}{2}$ as $n\to\infty$.

$\large\text{\underline{What are its properties?}}$

(Property 1)

The two consecutive terms are relatively prime.

(Proof)

Consider they are not relatively prime, or share at least one prime factor.

$\cdots\longrightarrow F_{n}=ak\cdots[1]$

$\cdots\longrightarrow F_{n+1}=bk\cdots[2]$

We know that,

$\cdots\longrightarrow F_{n+2}=F_{n}+F_{n+1}$.

So, by sum,

$\cdots\longrightarrow F_{n+2}=(a+b)k$.

We know that,

$\cdots\longrightarrow F_{n-1}=F_{n+1}-F_{n}$.

So, by subtraction,

$\cdots\longrightarrow F_{n-1}=(b-a)k$.

Now,

$\cdots\longrightarrow\text{ F_{n+1} and F_{n-1} are not relatively prime.}$

If we continue this process, we reach the following result which is

$\cdots\longrightarrow\text{ F_{n+1} and F_{1} are not relatively prime.}$

$\cdots\longrightarrow\text{ F_{n+1} and 1 are not relatively prime.}$

This leads to the contradiction. So, the opposite of our assumption is true, which means the consecutive terms of Fibonacci sequence are co-prime.

(Property 2)

$\cdots\longrightarrow\boxed{\displaystyle\sum^{n}_{k=1}F_{k}=F_{n+2}-1}$

(Proof)

We know that,

$\cdots\longrightarrow F_{n+2}=F_{n+1}-F_{n}$

The sum of the first n terms of the sequence is,

$\begin{aligned}\cdots\longrightarrow\displaystyle\sum^{n}_{k=1}F_{k}&=\sum^{n}_{k=0}(F_{k+2}-F_{k+1})\\&=(F_{3}-F_{2})+(F_{4}-F_{3})+\cdots+(F_{n+2}-F_{n+1})\\&=F_{n+2}-F_{2}\\&=F_{n+2}-1\end{aligned}$

Hence proven.

(Property 3)

$\cdots\longrightarrow\boxed{\displaystyle\sum^{n}_{k=1}F_{k}^{2}=F_{n}F_{n+1}}$

(Proof- Attachment)

Consider there are n squares which side lengths are $F_{1}$ to $F_{n}$. We can fill the rectangle which sides are $F_{n}$ and $F_{n+1}$, without leaving any square.

Hence shown.