Your very own repeating decimal

(10 pts)

Let’s build a repeating decimal based on your name as follows.

First, we’ll convert the letters of your name to digits by taking their position in the alphabet mod 10. You might think of “mod 10” as only caring about the digits in the one’s place. For example, my first name is “Mark” and the positions of those letters in the alphabet are

{M: 13, a: 1, r: 18, k: 11}

These yield the digits 3, 1, 8, and 1.

Now, write down the decimal expansion whose first few digits past decimal point are given by your first name and whose remaining digits are a repeating block of length two determined by your middle and last initial.

For example, my name is “Mark C McClure”. Thus, my number is

Problem:

Convert your very own repeating decimal to a fraction.

My Name’s numbers are:

{J: 10, o: 15, h: 8, n: 14}

So my digits are 0,5,8, and 4.
My middle name is Christian, so my final digit is 3.
Thus my number is 0.058433, with 33 repeating.

So my repeating decimal is:

I hope this is right, I was not sure if my 100000^n Term was correct. Also, I was not sure how to make the repeating bar on top of the 33.

My first name, middle and last initials are Audrey E. M. Using this web page you can see that my number is

Applying the geometric series to the sum, I see that

Thus, a rational representation of my own repeating decimal should be

My first name is as follows:

{M:13, a:1, g:7, g:7, i:9, e:5}

My middle name is Claire, so my first repeating digit is 3
My last name is Pasour, P is the 16th letter in the alphabet making my final repeating digit a 6

This means my number is

or

Applying the geometric series to the sum, I get

This means that my fraction is

My name is Bella with initials F and J so my decimal is 0.25221\overline{60}

I then used the formula \frac {a*r^n}{1-r} to solve for the sum:

So my decimal can now be represented by:

Rita L. R. - 18, 9, 20, 1, first repeating digit is 2, and second is 8 therefore- 0.8901\overline{28}. =

My first name is Thomas, so I have:

{T: 20, h: 8, o: 15, m: 13, a: 1, s: 19}

My middle and last initials are:
{P: 16, Y: 25}

all together, this makes my decimal:
0.085319\overline{65}

Now for the math:

My name is Sheldon W. Marshall giving me a decimal of

S:9, H:8, E:5, L:2, D:4, O:5, N:4, W:3, M:3

or 0.985245433

so

0.985245433 = \frac{9}{10}+\frac{8}{100}+\frac{5}{1000}+\frac{2}{10000}+\frac{4}{100000}+\frac{5}{1000000}+\frac{4}{10000000}+\frac{1}{10000000}\times \displaystyle \sum_{n=1}^{\infty} \frac{33}{100^n} =

=\frac{9852454}{10000000} +\frac{1}{10000000}\times \displaystyle \sum_{n=1}^{\infty} \frac{33}{100^n} =
=\frac{9852454}{10000000} +\frac{\frac{33}{100}}{1-(\frac{33}{100})}

=\frac{9852454}{10000000} +\frac{1}{10000000}\times\frac{33}{67} = \frac{769723}{781250}

My first name is Matt and according to the webpage my numbers are M: 3, A: 1, T: 0, T: 0
my middle initial is M: 3 and last name starts with a C: 3.
The number I am working with is

this number can be represented as

Applying the geometric series:

So, a rational representation of my very own repeating decimal is

My first name is Romulus, my middle initial is L and last name starts with an M. My decimal according to the corresponding numbers by alphabetical placement are;

0.8531219\overline{23}

This can be represented by;

0.8531219\overline{23}= \frac{8}{10}+\frac{5}{100}+\frac{3}{1000}+\frac{1}{10000}+\frac{2}{100000}+\frac{1}{1000000}+\frac{9}{10000000}+\frac{1}{10000000}\sum_{n=1}^{\infty}\frac{23}{100^n}

=\frac{8531219}{10000000}+\frac{1}{10000000}\sum_{n=1}^{\infty}\frac{23}{100^n}

Using geometric series we get,

\sum_{n=1}^{\infty}\frac{23}{100^n}=\frac{\frac{23}{100}}{1-\frac{23}{100}}=\frac{23}{77}

Altogether we can simplify to get the not so tidy fraction of;

My name is Andrew, my middle initial is J, and my last name starts with S. This leads to my decimal being:

0.144853\overline{19}

This can be stated as:

0.144853\overline{19}=\frac{1}{10}+\frac{4}{100}+\frac{4}{1000}+\frac{8}{10000}+\frac{5}{100000}+\frac{3}{1000000}+\frac{1}{1000000}\sum_{n=1}^{∞​}\frac{19}{100^n}

or

\frac{144853}{1000000}+\frac{1}{1000000}\sum_{n=1}^{∞​}\frac{19}{100^n}

Using geometric series I get:

\sum_{n=1}^{∞​}\frac{19}{100^n}=\frac{19/100}{1-19/100}=\frac{19}{99}

The rational representation of my own repeating decimal should be:

\frac{144853}{1000000}+\frac{1}{1000000}*\frac{19}{99}