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Math Help - [SOLVED] Sequence involving largest integer function.

  1. #1
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    [SOLVED] Sequence involving largest integer function.

    Let (a_n) be the sequence defined by a_n = [\frac{n^2 + 8n +10}{n+9}] where (x) is the largest integer that does not exceed x, Find the value of \sum_{n=1}^{30}a_n ??
    Last edited by mr fantastic; April 20th 2010 at 06:25 AM. Reason: Re-titled post.
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  2. #2
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    Hello wizard654zzz
    Quote Originally Posted by wizard654zzz View Post
    Let (a_n) be the sequence defined a_n = [\frac{n^2 + 8n +10}{n+9}] where (x) is the largest integer that does not exceed x, Find the value of \sum_{n=1}^{30}a_n ??
    Divide and get quotient and remainder:
    \frac{n^2 + 8n +10}{n+9}=(n-1) + \frac{19}{n+9}
    Now for 1\le n\le 10, \;2>\frac{19}{n+9}\ge 1. Hence a_n = n.

    But for n > 10, \;\frac{19}{n+9}<1. Hence a_n = n-1.

    So the answer is:
    1+2+3+...+10+10+11+12+...+29 = 445
    Grandad
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