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Some Examples of Mathematical Induction

Some Examples of Mathematical Induction. Example 1. Number of intersecting points. Given n lines on a plane. Suppose :. ( i) No two lines are parallel,. ( ii) No three lines are concurrent. Question : How many intersecting points ?. Some Examples of Mathematical Induction. When n = 4,

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Some Examples of Mathematical Induction

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  1. Some Examples of Mathematical Induction Example 1. Number of intersecting points Given n lines on a plane. Suppose : (i) No two lines are parallel, (ii) No three lines are concurrent. Question : How many intersecting points ?

  2. Some Examples of Mathematical Induction When n = 4, No. of intersecting points = 6 When n = 3, No. of intersecting points = 3 When n = 2, No. of intersecting points = 1

  3. Some Examples of Mathematical Induction Question : How many intersecting points ? From above, we have a table : n = 2 1 = (2)(1)/2 = (3)(2)/2 n = 3 3 = (4)(3)/2 n = 4 6

  4. Some Examples of Mathematical Induction Guess : For n lines, Question : How to prove it ? Answer : Try Mathematical Induction

  5. Some Examples of Mathematical Induction Idea : When n = 2, it is true. (direct checking) Assume that it is true when n = k.That is, there are k(k-1)/2 intersecting points. When n = k+1, we have (k+1) lines.You may view them as a system of k lines and one extra line is added on it.

  6. Some Examples of Mathematical Induction By induction hypothesis, the k lines have k (k-1)/2 intersecting points. Since (i) No two lines are parallel, and (ii) No three lines are concurrent The last line intersects the k lines at k distinct points. Totally we have [k(k-1)/2+k] intersecting points.That is, (k+1)(k)/2 intersecting points.

  7. Some Examples of Mathematical Induction By the principle of mathematical induction, the proposition is true for n = 2, 3, 4, ... That is, for n given lines,

  8. Some Examples of Mathematical Induction Example 2. Towers of Hanoi Towers of Hanoi is a classical game.

  9. Some Examples of Mathematical Induction You have a small collection of disks and three posts onto which you can put the disks. The disks all start on the leftmost post, and you want to move them to the rightmost post. The rules are as follows : (i) Never put a disk on top of a smaller one. (ii) The middle post is for intermediate storage.

  10. Some Examples of Mathematical Induction Question : What is the least number of steps for n discs ? Try it !

  11. Some Examples of Mathematical Induction From above, we have a table : n = 2 3 = 22 - 1 = 23 - 1 7 n = 3 = 24 - 1 n = 4 15

  12. Some Examples of Mathematical Induction Some Examples of Mathematical Induction Guess : For n discs, Again, how to prove it ? Answer : Try Mathematical Induction (again !)

  13. Some Examples of Mathematical Induction Idea : When n = 2, it is true. (direct checking) Assume that it is true when n = k.That is, we can move all the k discs to the leftmost post in 2k–1 steps. When n = k+1, we have (k+1) discs.We can first move the upper k discs to the middle post.It requires 2k–1 steps by induction hypothesis.

  14. Some Examples of Mathematical Induction Then we move the bottom (k+1)th discs to the leftmost post. This requires 1 step. Finally, we move the k discs in the middle post to the leftmost post. Again it requires 2k–1 steps by induction hypothesis. So total number of steps for k+1 discs = (2k–1) + 1 + (2k–1) = 2k + 2k – 1 = 2 (2k) – 1 = 2k+1– 1

  15. Some Examples of Mathematical Induction By the principle of mathematical induction, the proposition is true for n = 2, 3, 4, ... That is, for n discs,

  16. Some Examples of Mathematical Induction There is a legend about the Towers of Hanoi : At the beginning of time the priests in a temple were given a stack of 64 gold disks. Their assignment was to transfer the 64 disks from the leftmost post to the rightmost post. The priests worked very efficiently, day and night. When they finished their work, the myth said, the temple would crumble into dust and the world would vanish.

  17. Some Examples of Mathematical Induction By the above, The number of separate transfers of single disks the priests must make to transfer the tower is 264–1 = 18,446,744,073,709,551,615 moves! If the priests worked day and night, making one move every second it would take slightly more than 580 billion years (i.e. 580,000,000,000 years) to accomplish the job! It is estimated that the sun will run out of hydrogen fuel and this will result in the total destruction of the Earth after 4.5 billion years.

  18. Some Examples of Mathematical Induction You may download the game of Towers of Hanoifrom the following websites : http://sac-ftp.gratex.sk/educult31.html , or http://www.lhs.berkeley.edu/java/tower/towerhistory.html

  19. The End

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