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Thread: sequences

  1. #1
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    sequences

    Hi,

    $\displaystyle U_n=\sum_{i=0}^n \frac{1}{3^k}$

    $\displaystyle V_n=\sum_{i=0}^n \frac{k}{3^k}$

    How to show that: $\displaystyle 3V_{n+1}=V_n+U_n$????
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  2. #2
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    Hello bhitroofen01
    Quote Originally Posted by bhitroofen01 View Post
    Hi,

    $\displaystyle U_n=\sum_{i=0}^n \frac{1}{3^k}$

    $\displaystyle V_n=\sum_{i=0}^n \frac{k}{3^k}$

    How to show that: $\displaystyle 3V_{n+1}=V_n+U_n$????
    First, note that, when $\displaystyle k = 0, \frac{k}{3^k} = 0$. Therefore:
    $\displaystyle \sum_{k=1}^n \frac{k}{3^k}=\sum_{k=0}^n \frac{k}{3^k}=V_n$ ... (1)
    Then:
    $\displaystyle V_n+U_n = \sum_{k=0}^n \frac{k}{3^k}+\sum_{k=0}^n \frac{1}{3^k}$
    $\displaystyle =\sum_{k=0}^n \frac{k+1}{3^k}$

    $\displaystyle =3\sum_{k=0}^n \frac{k+1}{3^{k+1}}$

    $\displaystyle =3\sum_{k=1}^{n+1} \frac{k}{3^{k}}$

    $\displaystyle =3\sum_{k=0}^{n+1} \frac{k}{3^{k}}$, from (1)

    $\displaystyle =3V_{n+1}$
    Grandad
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