**Patterns I
found while iterating
functions/Chapter 8a**

by Maura, 10th grader

[This problem came from a teacher in one of Don's workshops.]

Mr. Cohen gave
me the function 5 + x/2 to iterate. I plugged in the number 10 for x and I got
the number 10 back out (5+10/2 = 5 + 5 = 10).
That was easy. Then I plugged **2** in for x. 5 + **2**/2 = 6.
Then I put the output number **6,** in for x ( 5 + **6**/2 = 8).
Then I put **8** in for x ( 5 + **8**/2 = 9).
So I got the infinite sequence** 2,
6, 8, 9, 9.5, 9.75, 9.875, 9.9375, 9.96875... **which
kept getting closer to 10. This putting back the
output number in for x repeatedly, is called feedback or
iteration. When I used the *Mathematica*^{' }command^{ }Take[N[NestList[f,2,200],300],-1]
, which tells the computer to do the iteration or feedback on our
function f , 200 times, carry it to 300 digits, and look at only the last number
in the sequence, we got:

9.99999999999999999999999999999999999999

9999999999999999999995021587777711086634

28474875697590060752779826502306629351910

67081682409327716932470528441414994291064

34426728355902014300227165222167968750000

00000000000000000000000000000000000000000

00000000000000000000000000000000000000000

0000000000000000
--very,
very close to **10**.

In
*Mathematica*^{'
}Mr. Cohen graphed this iteration using the command:

ListPlot[N[NestList[f,2,10],20],Prolog -> AbsolutePointSize[4]]

And
**every** number I started with to plug in for x, always formed an infinite
sequence that got closer and closer to **10**. I
noticed in the rule 5 + x/2, the numbers 5 and 2
and 5*2=**10**. When Mr. Cohen gave me the
function 6 + x/2, every number I plugged in for x
went to 12. I thought 5*2=**10** and 6*2=**12**,
I thought the rule was the number times the denominator. But when I used 6
+ x/3 I realized it didn't work. Every starting number goes to **9**.
So I kept working it out more and realized the numbers go to 6*3/2 or
6*3/(3-1). Using a calculator, then *Mathematica*^{'},
I found that every number I plugged in for x in

6
+ x/4 goes to **8 , **and
6*4/3
= 6*4/(4-1) = **8**

12
+ x/6 goes to **14.4, **and 12*6/5 =
12*6/(6-1) = **14.4**

13
+ x/3 goes to **19.5, **and 13*3/2 =
13*3/(3-1) = **19.5**

So
for the function a + x/b all numbers you start with
to plug in for x, will form an ** infinite sequence** that goes to **(****a*****b)/(b-1).**

**Fine
work, Maura!**

**Other
links to iterating functions**

Costa finds other patterns while iterating functions

10 ways to solve a quadratic equation, chapter 8a and answers to chapter 8a

Kelda

Iterate 6/x

Jamie
solves the quadratic equation x^{2}-x-1=0
to get x= Sqrt(x+1), then iterates the right
side to get the golden mean!

Mr Matsumoto wrote from Japan after reading Don's book in Japanese, to show how he gets the square root of a number by iteration

Finding
the sqrt of a number by iterating and averaging in
Chapter 10 **#6** and answers

Nanako's Mum solves a linear equation by iteration (map #22)

Jane iterates 2x (video)

Students
find the sqrt(sqrt(sqrt(...(x) what happens? See chapter
10 ** #4, Map upper left corner.**

Finding a new number- the Dottie number by iterating cos x

Finding an infinite continued fraction by iteration of 5 - 6/x

Graph
the iteration of **i ^{n}**

IES
applet iterates (a+bi)^{(a+bi)...} using Don's idea above in chapter
11

To other discoveries

To download Don's materials

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