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arithmetic

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The first term of a quadratic sequence is 8 and it's fourth is 32. If the second difference of the sequence is 2, find Tn.

  • arithmetic - ,

    the general term of a quadratic sequence is
    term(n) = an^2 + bn + c, where a, b, and c are constants
    It can be shown using differences that a is (1/2) of the second difference.
    Thus a = 1
    so term(1) = 8
    8 = 1(1^2) + b(1) + c ---> b+c = 7
    term(4) = 32
    32 = 1(2^2) + b(2) + c ---> 2b + c = 28

    subtract them: b = 21
    then c = 7-21 = -14

    term(n) = n^2 + 21n - 14

  • correction - arithmetic - ,

    I used the 32 as if it were the 2nd terms, should have been the 4th

    Here is the correct version


    Thus a = 1
    so term(1) = 8
    8 = 1(1^2) + b(1) + c ---> b+c = 7
    term(4) = 32
    32 = 1(4^2) + b(4) + c ---> 4b + c = 16

    subtract them:
    3b = 9
    b = 3

    then c = 7-3 = 4

    term(n) = n^2 + 3n + 4


    check:
    t(1) = 1+3+4 = 8
    t(4) = 16 + 12 + 4 = 32

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