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Math Help - sequences

  1. #1
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    sequences

    Hi,

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

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

    How to show that: 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,

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

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

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

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

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

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

    =3V_{n+1}
    Grandad
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