Backward reasoning - example
{"Image DocumentName": "Backward reasoning - example", "Image ImageDescription": "4/21/2017. Backward reasoning - example. Prove that the square of every odd integer has the form 8k + 1 for some integer k: Begin with some odd integer n, which by definition has the form n = 2i + 1 for some integer i. Then n2 = (2i + 1)2 = 4i2 + 4i + 1. We need to show that n2 has the form 8k + 1. Reasoning backwards, this follows if 4i2 + 4i can be written as 8k. But 4i2 + 4i = 4i(i + 1) i(i+ 1) is the product of 2 consecutive integers. Since every other integer is even, either i or i+1 is even. This means their product is even, and can be written 2k for some integer k. Therefore, 4i2 + 4i = 4i(i + 1) = 4(2k) = 8k; it follows that, since n2 = 4i2 + 4i + 1 and 4i2 + 4i = 8k, that n2 = 8k + 1. ch1.5&3.1.", "Image XResolution": "37", "Image YResolution": "37", "Image ResolutionUnit": "Pixels/Centimeter", "Image Software": "SlidePlayer", "Image YCbCrPositioning": "Centered"}