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Step 1 Number the first 25 lines on your paper, (1,2,3…)

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Step 2 Write any two whole numbers on the first two lines

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Step 3 Add the two numbers and write the sum on the third line

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Step 4 Add the last two numbers and write the sum on the next line

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Continue this process (add the last two, write the sum) until you have 25 numbers on your list).

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Select any number among the last five on your list and divide it by the number above it

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Remember I do not know your original two numbers or any of the 25 numbers on your sheet of paper So I cant know which of the last 5 numbers you have chosen to divide by the number above it

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Now I need to concentrate on the number presently shown on your calculator. If I close my eyes and think about your number, I will be able to prove to you that I know what your number will be.

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If you select any number between the last five (#21 to #25) and divides it by the number above it, the answer will always be1.618033989…, which just happens to be the Golden mean! (provided, of course, you have done all the addition correctly in steps 3-5 above)

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Its an incredible bit of mathematical trivia. Begin with any two whole numbers, make a Fibonacci-type addition list, take the ratio of two consecutive entries, and the ratio approaches the Golden Mean! The further out we go, the more accurate it becomes.

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Thats why we need 25 numbers to obtain sufficient accuracy. The proof requires familiarity with the Fibonacci Sequence, pages of algebra, and a knowledge of limits, all of which go far beyond the scope of explanation.

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If you divide one of your last five numbers by the next number (instead of the previous number), the result is the same decimal without the leading 1.

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AB Euclid of Alexandria (325 – 265 BC.) In Book VI of the Elements, Euclid defined the "extreme and mean ratios" on a line segment. He wished to find the point (P) on line segment AB such that, the small segment is to the large segment as the large segment is to the whole segment. P In other words how far along the line is P such that: We use a different approach to Euclid and use algebra to help us find this ratio, however the method is essentially the same. 1 Let AP be of unit length and PB =.Then we require such that Solving this quadratic and taking the positive root. We get the irrational number shown. is the Greek letter Phi.

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Fibonacci Sequence 1716 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 33 464 5510 6156 20 7 721 35 88 2856285670 9 3684 126 84 36 9 10 45 210 252 210 120 45 120 10 11 55 330 462 330 165 55 11 12 66 495 792924 792 495 220 66 12 13 78 286 715 1287 1716 1287 715 286 78 13 Add the numbers shown along each of the shallow diagonals to find another well known sequence of numbers. 1 23813 21 5589 15 34 144 233 377 The sequence first appears as a recreational maths problem about the growth in population of rabbits in book 3 of his famous work, Liber – abaci (the book of the calculator ). Fibonacci travelled extensively throughout the Middle East and elsewhere. He strongly recommended that Europeans adopt the Indo-Arabic system of numerals including the use of a symbol for zero zephirum The Fibonacci Sequence Leonardo of Pisa 1180 - 1250

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