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Pi in Pascal's Triangle (2014)
- lixtra 1y ago(2014) [1] [1] https://web.archive.org/web/20140901000000*/https://www.cut-the-knot.org/arithmetic/algebra/PiInPascal.shtml https://web.archive.org/web/20140901000000*/https://www.cut-...
- zkmon 1y agoI wonder if the triangle hides any secrets related to prime numbers as well.
- dhosek 1y 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 1y 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 1y agoThat requires you to prove the binomial theorem first, though, and won't that need induction?
- dhosek 1y agoDepends on your starting point.
- adornKey 1y agoIndeed it does, e.g: https://nonagon.org/ExLibris/paul-erdos-bertrands-conjecture https://nonagon.org/ExLibris/paul-erdos-bertrands-conjecture
- JohnKemeny 1y 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 1y ago0th triangle number and I think that can probably be connected to 1+1-1+1-1+⋯=½ somehow.
- deleted 1y ago[deleted]
- nimonian 1y agoThere's a typo in the final line of math. I think 1/7C2 should be positive rather than negative.
- raldi 1y agoEven after reading this, I don't understand what pi has to do with 4, 20, 56, 120, 220, 364
- kevmo314 1y agoIt's the equation in the diagram: π = 3 + 2/3 (1/4 − 1/20 + 1/56 - ...)