k, if S (k) assertion is true, follows the verity of the assertion S (k +1), then the assertion S (n) is true for all the natural n "in geometric way means the following transition:
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In that way we get a belt where the first two squares are colored:
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By repeating the transition one more time, we get a belt where the first three squares are colored:
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Consequently, if we continue in the same way, then gradually all the infinite belt will be colored and the general assertion n will be proved. One of the basic schemes of mathematical induction is Weak Mathematical Induction: Let S (n) be a statement involving n. If S (1) holds, and for every k? 1, S (k) В® S (k +1), then for every n? 1, the statement S (n) holds. This can be depicted as follows:
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For example, Task 3: Prove that for n? 2, 4n 2 > n + 11.induction scheme is Strong Mathematical Induction: Let S (n) denote a statement involving an integer n. If S (k) is true and for every m? k, S (k) Г™ S (k +1) Г™ ... Г™ S (m) В® S (m +1) then for every n? k, the statement S (n) is true. This can be depicted as follows:
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For example, Task 4: Prove that a n = 5 . 2 n - 3 n +1 , if a 1 = 1 , a 2 = -7 and a n +2 = 5a n +1 - 6a n for all n? 1.another induction scheme is Downward Mathematical Induction: Let S (n) be a statement involving n. If S (n) is true for infinitely many n, and for each m? 2, S (m) В® S (m-1) then for every n? 1, the statement S (n) is true. Its graphical depiction is:
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For example, Task 5: Prove that the statement "the geometric mean of n positive numbers is not larger than the arithmetic mean of the same numbers" is true, ie,
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At schools, teaching the method of mathematical induction, usually the simplest schemes are covered however more compli...
