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7 February, 07:48

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2 - n + 2

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  1. 7 February, 08:02
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    Let n = 1

    then f (1) = 1^1 - 1 + 2 = 2 so it is true for n = 1

    for the next number after n (n+1) we have f (n+1) =

    (n+1) ^2 - (n+1) + 2

    = n^2 + 2n + 1 - n - 1 + 2

    = n^2 + n + 2

    = n (n+1) + 2

    Now n (n+1) must be divisible by 2 because either n is odd and n+1 is even OR n is even and n+1 is odd and odd & even always = an even number.

    So the function is divisible by 2 for n+1 We have shown that its true for n = 1 Therefore it must be true for n = 1,2,3,4 ...

    True for all positive integers
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