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Math Help - Next number in sequence problem/solving system of linear equations.

  1. #1
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    Next number in sequence problem/solving system of linear equations.

    0 12 10 0 -12 -20

    I calculate the differences, and find it's a 3rd degree polynomial.

    An^3+Bn^2+Cn+D

    So I know when n=1 the above equation=0, and when n=2, 12, and so on.

    So I come up with these 4 equations

    a + b+ c+d=0
    8a+ 4b+2c+d=12
    27a+ 9b+3c+d=10
    64a+16b+4c+d=0

    Now I have to solve this system of linear equations. I just keep trying and getting wrong answers. What's the best way to attack this problem?
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  2. #2
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    Re: Next number in sequence problem/solving system of linear equations.

    Use technology, continual substitution over a system of this order leaves too much room for error.
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  3. #3
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    Re: Next number in sequence problem/solving system of linear equations.

    Quote Originally Posted by StudentMCCS View Post
    a + b+ c+d=0 [1]
    8a+ 4b+2c+d=12 [2]
    27a+ 9b+3c+d=10 [3]
    64a+16b+4c+d=0 [4]
    Do subtractions: [2]-[1], [3]-[1] and [4]-[1]; will leave
    7a + 3b + c = 12 [5]
    26a + 8b + 2c = 10 [6]
    63a + 15b + 3c = 0 [7]

    Divide [6] by 2 and [7] by 3:
    7a + 3b + c = 12 [5]
    13a + 4b + c = 5 [6]
    21a + 5b + c = 0 [7]

    Get it? Take over...
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  4. #4
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    Re: Next number in sequence problem/solving system of linear equations.

    Since you have already calculated the differences, and seen that the "third differences" are all the same, I would suggest using "Newton's divided difference formula". The value and differences "at 0" are 0, 12, -14, 6 so the polynomial is
    0+ 12x- (14/2!)x(x- 1)+ (6/3!)x(x-1)(x-2)= 12x- 7x(x- 1)+ x(x-1)(x-2) = 12x- 7x^2+ 7x+ x^3- 3x^2+ 2x= x^3- 10x^2+ 21x
    Check: 0^3- 10(0^2)+ 21(0)= 0, [itex]1^3- 10(1^2)+ 21(1)= 12[/tex], 2^3- 10(2^2)+ 21(2)= 10[/itex], [itex]3^3- 10(3^2)+ 21(3)= 0, 4^3- 10(4^2)+ 21(4)= -12, 5^3- 10(5^2)+ 21(5)= -20.
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  5. #5
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    Re: Next number in sequence problem/solving system of linear equations.

    using matrices will be helpful for solving such equations
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  6. #6
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    Re: Next number in sequence problem/solving system of linear equations.

    Thanks everyone.
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