Pi in Pascal's Triangle (2014)
68 points by senfiaj 1 month ago | 14 comments- JohnKemeny 1 month agoπ - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••
That's beautiful. I wonder why the -2 is there, though. To fix it, we would need
π = -x + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••
where x = 2, and so it would be
π = -1/½ + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••
which makes the -1st triangle number ½, I guess.
- dhosek 1 month ago0th triangle number and I think that can probably be connected to 1+1-1+1-1+⋯=½ somehow.
- dhosek 1 month ago
- raldi 1 month agoEven after reading this, I don't understand what pi has to do with 4, 20, 56, 120, 220, 364
- kevmo314 1 month agoIt's the equation in the diagram:
π = 3 + 2/3 (1/4 − 1/20 + 1/56 - ...)
- kevmo314 1 month ago
- lixtra 1 month ago
- nimonian 1 month agoThere's a typo in the final line of math. I think 1/7C2 should be positive rather than negative.
- zkmon 1 month agoI wonder if the triangle hides any secrets related to prime numbers as well.
- dhosek 1 month agoIt’s possible, I suppose.
One of my favorite proofs that the sums of each row are powers of two comes from the fact that the numbers in row n+1 are the coefficients of the powers of (a+b)ⁿ, so setting a=b=1 you get 2ⁿ (most discrete math students seeking to prove this end up reaching for induction which is a heavier proof than this).
- nimonian 1 month agoI like the argument that every number in the row below is formed by summing two numbers from above. So each number above appears twice below. Hence the sum doubles.
- madcaptenor 1 month agoThat requires you to prove the binomial theorem first, though, and won't that need induction?
- dhosek 1 month agoDepends on your starting point.
- dhosek 1 month ago
- nimonian 1 month ago
- adornKey 1 month agoIndeed it does, e.g:
https://nonagon.org/ExLibris/paul-erdos-bertrands-conjecture
- dhosek 1 month ago
- 1 month ago