SENIOR_1.4

Date	Activities / Objectives	Images / Home work / Assignments	Remarks / Images	References / Links
Friday April 1st 11:25 -12:05	Return of Exercise 2.3 / Recursive sequence with an arithmetic auxiliairy sequence to prove the limit. Presentation of the problem of the Hanoï towers : construction of the recursive formula : M_n = 2M_n-1+ 1 where M_n = represents the minimum number of moves. Demo of the direct formula M_n = 2ⁿ - 1 Problem of the Chess Board and the amount of rice to be stacked on each box by doubling the previous quantity : The total number of rice grains on the 64^th box would be 2⁶³ and the total number of rice grains on the board would be equal to M₆₄ = 2⁶⁴ - 1 (same as the number of moves in the Hanoï towers) Study of the general formula of the sum of the first n^th terms of a geometric sequence : (if q ≠ 1) Study of the limits of S_n with \|q\| < 1 the limit of qⁿ = 0 when \|q\| < 1 and n-> ∞	Game of the Hanoï Towers M_n = M_n-1+ 1 + M_n-1M_n + 1= 2M_n-1+ 2 = 2(M_n-1+ 1) M_n+ 1 = 2ⁿ * * * How to make 2 candy bars with 1 : (April's 1st Fool's day ...) Take a candy bar, chop it up in two equal parts and each new piece in two ... indefinitely, then glue them back together ... on top of the first half, then ... you get TWO candy bars instead of one ...	The puzzle was invented by the French mathematician Édouard Lucas in 1883. There is a legend about an Indian temple which contains a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the rules of the puzzle, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle is completed, the world will end.^[1] It is not clear whether Lucas invented this legend or was inspired by it. If the legend were true, and if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 2⁶⁴−1 seconds or roughly 585 billion years;^[2] it would take 18,446,744,073,709,551,615 turns to finish ... Similarly see Problem of the Chess Board there is a legend about the quantity of rice grains represented by this number would cover more than three times the whole surface of the Earth	Game of the Hanoï Towers About the chess board and the Formula of S_n for a Geometric sequence : see 高中数学＃5 － B - 2005 p. 51-52 Problem of the Chess Board When the creator of the game of chess (in some tellings an ancient Indian mathematician, in others a legendary dravida vellalar named Sessa or Sissa) showed his invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention. The man, who was very wise, asked the king this: that for the first square of the chess board, he would receive one grain of wheat (in some tellings, rice), two for the second one, four on the third one, and so forth, doubling the amount each time. The ruler, arithmetically unaware, quickly accepted the inventor's offer, even getting offended by his perceived notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the wheat to the inventor. However, when the treasurer took more than a week to calculate the amount of wheat, the ruler asked him for a reason for his tardiness. The treasurer then gave him the result of the calculation, and explained that it would be impossible to give the inventor the reward. The ruler then, to get back at the inventor who tried to outsmart him, told the inventor that in order for him to receive his reward, he was to count every single grain that was given to him, in order to make sure that the ruler was not stealing from him.
Wednesday April 6 11:25 -12:05	Presentation of the Fibonacci's sequence F_n+1= F_n + F_n-1 ; F₁ = F₂ = 1 Exercises on the Fibonacci's numbers and it's relationship with the Golden number.	Preparation of TEST of Friday April 8th Review all previous exercises 1.1-1.2-2.1-2.2-2.3-2.4 about recursive or non recursive sequences + formulas of the Geometric series Dowload the .ppt prepared by 吴鹏老师 for his visit to my class at Ecole alsacienne in January 2009	Fibonacci (XIII^th c.)	Review : Arith. & Geom. Sequences Memo Ex. 1.1 & 1.2 ANSWERS Ex. 2.1 & 2.2 ANSWERS Ex. # 2.3 ANSWERS Ex. # 2.4 The Chaos Theory 吴鹏老师.ppt Exercises on Fibonacci's # 2.5
Friday April 8 11:25 -12:05	TEST about recursive sequences			Test A Test B
Wednesday April 13 11:25 -12:05	Return graded Tests Tests Answers		TESTS RESULTS (maximum score : 40 pts) 32 ≤ N ≤ 40 := 10% 28 ≤ N ≤ 31 := 20% 24 ≤ N ≤ 27 := 22% 00 ≤ N ≤ 23 := 48%
Friday April 15 11:25 -12:05	Geometric Series in ... Geometry (I) Study of the finite Length of an Infinite Spiral			Geometric Series in ... Geometry (I)
Wednesday April 20 11:25 -12:05	Geometric Series in ... Geometry (II)		Comparison of the inifinite length of the perimeter with the finite measure of the area	Geometric Series in ... Geometry (II) Koch Snowflake Answers
Friday April 22 11:25 -12:05	Geometric Series in ... Geometry (III)			Geometric Series in ... Geometry (III) Riemann's Integral Sum ANSWERS
Wednesday April 27 11:25 -12:05	Mid-Term Exams (general - Non elective subjects)
Friday April 29 11:25 -12:05	Mid-Term Exams (general - Non elective subjects)