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Download Lecture 1: Worksheet Triangular numbers 1 3 6 10 15 21 36 45
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Math 1A: introduction to functions and calculus Oliver Knill, 2012 Tetrahedral numbers Lecture 1: Worksheet In this first lecture, we want to see that the essence of calculus is already in basic arithmetic. We stack now spheres onto each other building n layers and count the number of spheres. The number sequence we get are called tetrahedral numbers. Triangular numbers 1 4 10 20 35 56 84 120 ... We stack disks onto each other building n layers and count the number of discs. The number sequence we get are called triangular numbers. 1 3 6 10 15 21 36 45 ... n=1 n=2 n=3 Also this sequence defines a function. For example, g(3) = 10. 2 Verify that n(n − 1)(n − 2) 6 satisfies g(n + 1) − g(n) = n(n − 1)/2. g(n) = n=4 This sequence defines a function on the natural numbers. For example, f (4) = 10. 1 n=1 n=2 You have just verified the formula Verify that n(n − 1) 2 gives the above numbers. Check this by algebraically evaluating f (n) = Df (n) = f (n + 1) − f (n) . Carl-Friedrich Gauss, 1777-1855 n=3 n=4 d n [x] = n[x]n−1 dx in the case n = 1, 2, 3, where Df (n) = f (n + 1) = f (n) is the difference and [x]n = x(x − 1)(x − 2) . . . (x − n + 1) is the ”quantum power”. In the homework, you push this a bit further.