Pythagoras Unchained To identify patterns, we looked at the first and second order differences in the sequences. Below is sequence 1. Sequence 1: 1 2 5 10 17 26 First Order Difference: 1 3 5 7 9 Second Order Difference: 2 2 2 2 In sequence 2, the first and second order differences are identical to those for sequence 1: Sequence 2: 4 5 8 13 20 29 First Order Difference: 1 3 5 7 9 Second Order Difference: 2 2 2 2 Likewise, in sequence 3: Sequence 3: 9 10 13 18 25 34 First Order Difference: 1 3 5 7 9 Second Order Difference: 2 2 2 2 Also, sequence 4: Sequence 4: 16 17 20 25 32 41 First Order Difference: 1 3 5 7 9 Second Order Difference: 2 2 2 2 When the second order difference of a sequence is constant, one knows the sequence can be produced by a quadratic function. That is, there exist a, b, and c such that, in sequence 1, a ⋅12 + b ⋅1 + c =1 , a ⋅ 22 + b ⋅ 2 + c =, 2 a ⋅ 32 + b ⋅ 3 + c =, 5 and so on. Because there are 3 unknowns, 3 equations are sufficient to find the values of a, b, and c. This leads to the following system of equations for sequence 1: 1a + 1b + c = 1 4a + 2b + c = 2 which, written as an augmented matrix, is 9a + 3b + c = 5 The row reduced echelon form of this matrix is 1 1 1 1 4 2 1 2 9 3 1 5 1 0 0 1 0 1 0 −2 0 0 1 2 That is, a = 1, b = −2, and c = 2 . Therefore, the function f1 ( n ) = n 2 − 2n + 2 will produce sequence 1. As n takes on the values 1, 2, 3, 4, ..., f1 ( n ) takes on the values 1, 2, 5, 10, ... which are the radicands of the lengths of segments in column 1. 28 | Spring/Summer 2018 Texas Mathematics Teacher

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