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Math

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Use Mathematical Induction to show that the statement
2 + 6 + 10 + . . . + (4n – 2) = 2n^2
is true

  • Math - ,

    n an sum n
    1 2 2
    2 6 8
    3 10 18
    4 14 32

    n (4n-2) 2n^2

    (n+1) (4(n+1)-2) [2n^2+ (4(n+1)-2)] or hopefully 2(n+1)^2

    [2n^2+ (4(n+1)-2)] = 2n^2+4n +2
    and
    2(n+1)^2 = 2(n^2+2n+1) = 2n^2+4n+2 done

    2n^2+ (4(n+1)-2)

  • Math - ,

    Using the 3 step process:
    1. test for n=1
    LS = 2
    RS = 2(1^2) = 2

    2. assume it true for n=k
    that is ....
    2+4+6 + ... + (4k-2) = 2k^2

    3 . prove it is then true for =k+1
    or
    2+4+6+ ... + (4k-2) + 4(k+1)-2 = 2(k+1)2
    LS = [2+4+6+...+ 4k-2 ] + 4(k+1)-2
    = 2k^2 + 4k + 4 - 2
    = 2(k^2 + 2k + 1)
    = 2(k+1)^2
    = RS

    QED

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