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Nitya Nigam Pascal's triangle, as many of you may already know, is a mathematical pattern where the nth row is the sum of nCk from k=0 to k=n. As shown in the diagram on the left, each number is obtained by adding together the two numbers directly above it. In this article, we will investigate some of the patterns hidden in Pascal's triangle. As mentioned in the solution to May 24th's problem of the week, the sum of nCk from k=0 to k=n is 2^n, which means that the sum of rows in Pascal's triangle is 2^n. We can prove this quite easily using the binomial theorem. The binomial theorem states that: Letting x=y=1: which proves our claim. Another interesting pattern in the triangle is that the squares of the numbers on the second diagonal are equal to the sum of the two numbers directly next to and below it, as shown in the diagram below: To prove this, we need to recognise that the third diagonal is the triangular numbers, which have formula n(n+1)/2. The sum of two consecutive triangular numbers is: which is the necessary square number.
Pascal's triangle is full of many more interesting patterns - let us know in the comments what you'd like to learn about next!
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