Sequences and Series

Lesson

The Fibonacci sequence is named after the 13th Century mathematician Leonardo de Fibonacci of Pisa Italy. It is a sequence that appears in nature - the branching of trees, the family tree of the honey-bee, the count of petals on flowers and many other instances.

Let's begin by looking at how to create a Fibonacci Spiral of our own.

**Start with a blank piece of graph paper, and highlight a square with side length 1, somewhere towards the middle. **

**Next to that, highlight another square of side length 1.**

**Now, highlight a square of side length 2. Use the two squares from before as one of the sides. (like this image).**

** **

Next, build a square of length 3 on the side of the squares we have already built, then a side of length 5 and another of length 8. It should start to look like what I have here.

The final step is draw in a the spiral. We construct the spiral but joining diagonally opposite corners in a swooping arc motion. Start in the centre with the first square and then join corner to corner. This may take a little bit of practice to get the arcs looking nice.

And so you have constructed a Fibonacci spiral.

The Fibonacci sequence is $1,1,2,3,5,8,13,21,34,55,89,...$1,1,2,3,5,8,13,21,34,55,89,... where, apart from the first two terms, each term is the sum of the previous two terms. It also possesses some unusual mathematical properties such as:

- The sum of the squares of any two consecutive terms is another Fibonacci term.
- The ratio of consecutive terms, namely $\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8}...$32,53,85,138... approaches the golden ratio where $=$= $\frac{1+\sqrt{5}}{2}=1.618033...$1+√52=1.618033...

- Apart from $F_1$
`F`1, $F_2$`F`2 and $F_4$`F`4 every prime term is sitting in a prime position. That is, if the $n$`n`th term $F_n$`F``n` is prime then $n$`n`must be prime.

As an aside, the Fibonacci sequence is a member of a more general family of sequences which are all defined by how many terms are added to produce each new term. For example, the *Tribonacci* sequence begins $1,1,1,3,5,9,17,31,...$1,1,1,3,5,9,17,31,... where apart from the first three terms, each new term is formed as the sum of the previous three terms. The *Tetranacci* sequence begins $1,1,1,1,4,7,13,25,49,94,..$1,1,1,1,4,7,13,25,49,94,.. where the four previous terms are added. It is interesting to note that the ratio of successive terms approaches a unique limiting value given by the real positive solution to a sequence of equations. For the Fibonacci sequence, we can show that the limiting value is phi given by the positive solution to the equation $x^2-x-1=0$`x`2−`x`−1=0. Similarly, the limiting value for the Tribonacci sequence is given by the positive solution to $x^3-x^2-x-1=0$`x`3−`x`2−`x`−1=0. For the Tetranacci sequence the limiting value is given by the positive solution to $x^4-x^3-x^2-x-1=0$`x`4−`x`3−`x`2−`x`−1=0. The pattern continues indefinitely with all limiting values forming their own sequence of numbers that approach but never exceed $2$2.

We can state the Fibonacci sequence using the recurrence relationship defined as:

$F_{n+2}=F_n+F_{n+1}$`F``n`+2=`F``n`+`F``n`+1 with the first two terms $F_1=1$`F`1=1,$F_2=1$`F`2=1

The rule for the $n$`n`th term is far more difficult to find, but in case you're curious, it is given by:

Consider the following sequence.

$2,-1,1,0,1,1,\text{. . .}$2,−1,1,0,1,1,. . .

Is the sequence a Fibonacci-type sequence (where each term is the sum of the two preceding terms)?

Yes

ANo

BYes

ANo

BWhat are the next two terms of the sequence?

Write both terms on the same line, separated by a comma.

Use the fact that the Fibonacci sequence is defined by $t_n=t_{n-2}+t_{n-1}$`t``n`=`t``n`−2+`t``n`−1, where $t_1=1$`t`1=1 and $t_2=1$`t`2=1, to generate terms $3$3 to $10$10.

Write all the values on the same line, separated by commas.

In the Fibonacci sequence, $t_{21}=10946$`t`21=10946, $t_{23}=28657$`t`23=28657 and $t_{24}=46368$`t`24=46368.

Find $t_{25}$

`t`25.Find $t_{19}+t_{20}$

`t`19+`t`20.Find $t_{22}$

`t`22.

Use arithmetic and geometric sequences and series

Apply sequences and series in solving problems