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20 August, 11:24

1. Use strong induction to show that if you can run one mile or two miles, and if you can always run two more miles once you have run a specified number of miles, then you can run any number of miles.

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  1. 20 August, 11:43
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    Proof

    Step-by-step explanation:

    Given:

    - If you run 1 or 2 miles

    - You can always run 2 miles more after running specified number of miles

    Find:

    - Prove that you can run any number of miles

    Solution:

    - Let M (n) be " You can run the nth mile"

    Basis step: n = 1 and n = 2

    - M (1) and M (2) are True, because you can run one or two miles as given in statement.

    Inductive Step:

    - We assume that M (1), M (2), ..., M (k) are all true, thus you can run the first k miles.

    - We then need to prove that M (k + 1) is also true.

    - Since M (k - 1) is true then M (k + 1) is true. (You can always run 2 miles more after running specified number of miles)

    Conclusion:

    - By the principle of strong induction, M (n) is true for all positive n integers.
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