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Math Help - recurrence

  1. #1
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    recurrence

    the range is given in this way:
    y1=-1
    y(n+1)=1/3y(n)+3

    Find a f function y(n)=f(n)

    thanks
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  2. #2
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    Quote Originally Posted by blertta View Post
    the range is given in this way:
    y1=-1
    y(n+1)=1/3y(n)+3

    Find a f function y(n)=f(n)

    thanks
    For y_{n+1} = \frac{1}{3} y_n + 3 for let y_n = z_n + k so the difference equation becomes

    z_{n+1} + k = z_n + \frac{1}{3}\left( z_n + k \right) + 3

    and choosing  k = \frac{9}{2} gives

    z_{n+1} = \frac{1}{3} z_n

    Seeking a solution of this in the form

    z_n = c \rho^n gives \rho = \frac{1}{3}, thus giving the solution

    y_n = c \left(\frac{1}{3}\right)^n + \frac{9}{2}

    Now use your initial condition to find c.

    Note. If the difference equation is y_{n+1} = \frac{1}{3 y_n} + 3, it's harder problem.
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