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Pick any positive real number (like 10 or pi or .78 or whatever), add 1, take the square root, add 1, take the square root, add 1, take the square root, etc. forever. What happens? Try it on your calculator, and you will see that it gets closer and closer to some number. Here I started with 10 using my calculator (The first number on each line is the square root of the previous number):
On my calculator, it doesn't change after that. That number may look familiar to you. That is the Golden Ratio. What a coincidence, huh? Well, not really. We kept taking square roots. For large numbers (greater than one), the square root reduces the number. For small numbers (less than one), the square root increases the number. So we should expect that the above process zeroes in on a region of numbers close to one. It should not surprise us that it zeroes in on a particular number. What particular number would it zero in on? How about a number that is one less than its square? After all, we "add 1, take the square root, add 1, take the square root, add 1, take the square root..." This process will not change when the number is one less than its square. And that is one of the features of the golden ratio. If the above process does settle down to one specific number, then it should settle down to the golden ratio.