Golden Proof - Numberphile
Part Two of Golden Ratio Trilogy Proof that an infinite number of sequences have that "golden ratio" property - not just the Fibonacci Numbers. More links & stuff in full description below...
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The Fibonacci numbers again start:1, 1, 2, 3, 5and the rule is: if each number hereequals the one before itSo, if this was the nth numberand we're gonna call that x nfor the nth numberit equals the one before itplus the one before thatso that equalsx n minus oneplus x n minus twoSo that's our finding thingand what we're trying to findultimatelyis what is the ratio between the two of the numbersand importantly, becausewe're looking for the ratioit is approaching the limitit doesn't matter which two termsit should be the same ratio, give or takeandif you do infinitely many the same ratio for all of thembut, rrrrhhhhhSo. Uh. For any two of the termsso what we can say nowis the ratio betweenx to the n and the one before thatwhich was ...
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The Fibonacci numbers again start:1, 1, 2, 3, 5and the rule is: if each number hereequals the one before itSo, if this was the nth numberand we're gonna call that x nfor the nth numberit equals the one before itplus the one before thatso that equalsx n minus oneplus x n minus twoSo that's our finding thingand what we're trying to findultimatelyis what is the ratio between the two of the numbersand importantly, becausewe're looking for the ratioit is approaching the limitit doesn't matter which two termsit should be the same ratio, give or takeandif you do infinitely many the same ratio for all of thembut, rrrrhhhhhSo. Uh. For any two of the termsso what we can say nowis the ratio betweenx to the n and the one before thatwhich was ...
