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How would I calculate the sum or area of all the numbers in Pascal's Triangle stopping at row 111?

Would I refer to this as the sum? Or the area (A) within the triangle?

Also, how would I create a graphical image of Pascal's Triangle in Base 111? Like this Pascal's Triangle and how do I express a number that is within the row 111?

I have invested hours of research into these questions. The mathematical literature sends me in an infinite rabbit hole of confusion. I tend to struggle with basics.

Thank you for your knowledge if you've shared some with me.

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    $\begingroup$ Since the sum of the numbers in the $n$-th row is $2^n$, and the sum of the numbers in the first $n$ rows is $[2^{(n+1)} - 2]$, are you asking how to convert $[2^{(n+1)} - 2]$ from base $(10)$ to base $(111)$? $\endgroup$
    – user2661923
    Commented Feb 2, 2021 at 2:08
  • $\begingroup$ I guess i am asking what is the sum of all 111 rows and each of the individual row totals. This answer will give me full scope of thy project. If I could see 111 rows in an image that would be ideal. $\endgroup$
    – mister mcdoogle
    Commented Feb 2, 2021 at 4:50
  • $\begingroup$ Clarifications requested: [1] "...If I could see 111 rows in an image..." : What image? [2] "...what is the sum of all 111 rows" : All 111 rows ?? Pascal's triangle has an infinite number of rows. [3] Beyond points [1] and [2] above, since my previous comment enumerates each of the 1st 111 rows, and the sum of the 1st 111 rows, is your actual question, (whatever that may be) answered? $\endgroup$
    – user2661923
    Commented Feb 2, 2021 at 9:49
  • $\begingroup$ Base $16$ uses $16$ digits $0,1,2,3\ldots,A,B,C,D,E,F$. Base $111$ needs $111$ differents digits. Is this what you really want? $\endgroup$
    – Raffaele
    Commented Feb 2, 2021 at 17:48
  • $\begingroup$ Yes Base 111 is what I want. I will go over all this information soon. Trying to communicate as best as I can here. $\endgroup$
    – mister mcdoogle
    Commented Feb 2, 2021 at 19:12

1 Answer 1

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Since the formula for calculating the sum of the numbers in row $n$ is $2^n$, the sum of all numbers up to a certain point is the sum of the first $n$ $2^n$ numbers, or in other case, $$2^{n+1} -1$$ In your case with the row being up to 111, the steps to answer this question is:

Step 1: Substitute $n$ with 111 $$2^{111+1} -1$$

Step 2: Use a calculator to calculate the expression because the product is very large.

The answer is $$5.192296858... * 10^{33}$$

Remember to calculate the "Area" of a pascal triangle up to base $n$ is $2^{n+1} -1$ :)

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