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Thread: Recurrence Sequences

  1. #1
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    Recurrence Sequences

    A sequence of terms $\displaystyle {U_n}$ is defined by $\displaystyle n\geq1$ by the recurrence relation $\displaystyle U_{n+1}=kU_n+2$, where $\displaystyle k$ is a constant. Given that $\displaystyle U_1=3$:

    a) Find an expression in terms of $\displaystyle k$ for $\displaystyle U_2$.


    Please tell me how to do this .
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  2. #2
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    Quote Originally Posted by Viral View Post
    A sequence of terms $\displaystyle {U_n}$ is defined by $\displaystyle n\geq1$ by the recurrence relation $\displaystyle U_{n+1}=kU_n+2$, where $\displaystyle k$ is a constant. Given that $\displaystyle U_1=3$:

    a) Find an expression in terms of $\displaystyle k$ for $\displaystyle U_2$.


    Please tell me how to do this .
    Let $\displaystyle n = 1$ in $\displaystyle U_{n+1}=kU_n+2$, so

    $\displaystyle U_{2}=k U_1+2 = 3k + 2$.
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    part b says "hence find an expression for $\displaystyle U_3$". The answer is $\displaystyle 3k^2+2k+2$ but I have no idea how to get this answer. Do you know of any notes on the internet with recurrence sequences?
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    Quote Originally Posted by Viral View Post
    part b says "hence find an expression for $\displaystyle U_3$". The answer is $\displaystyle 3k^2+2k+2$ but I have no idea how to get this answer. Do you know of any notes on the internet with recurrence sequences?
    Since $\displaystyle U_{n+1} = k U_n +2$ then substitute $\displaystyle n = 2$ so

    $\displaystyle U_3 = kU_2 + 2$ but you already know $\displaystyle U_2 = 3k +2$ from the first part.

    You might try googleing it on the web. Like I did to get this site

    Difference Equation Tutorial

    Try the word "difference equation".
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