Question

The Arithmetical Triangle sparalleles arta Blaise Pascal's 1955 work Treatise on the Arithmetical Triangle contains a collection several results already known about the "Pascal's Triangle" for more than 500 years, as well as applications of the triangle to probability. Among these results are the Hockey-Stick Theorem the fact that the sum of a row is a power of 2, as well as the following fact about ratios: の(k+1) = (k + 1):(n-k) We have actually already seen this fact somewhere else, although one new thing here is that Pascal's definition of the triangle was purely arithmetic (not about counting) So he defined terms as the sum of the two terms above. 1. Verify the ratio above for a few examples. 2. Suppose the ratio holds for n 4 and all values of k. Demonstrate how you can use this to show that the ratio words for some cases of n step by example 5. That is, do part of the induction 3. Give a full proof by induction (on n for the fact 4. Assuming the fact is true, how does this give you a formula to compute ()?

Answer 1

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5-k Sck t _1Hols

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