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Thread: tricky!

  1. #1
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    Smile tricky!

    help,,, the geom is tricky

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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello aeroflix

    Here's the solution to number 1.

    Suppose we take $\displaystyle n$ natural numbers starting from $\displaystyle m$. Then their sum is:
    $\displaystyle S = \tfrac{1}{2}n(2m+n-1)$
    $\displaystyle =1000$
    $\displaystyle \Rightarrow n(2m+n-1) = 2000$

    $\displaystyle \Rightarrow m = \frac12\left(\frac{2000}{n}-n+1\right)$
    $\displaystyle \Rightarrow n$ is a factor of $\displaystyle 2000$ and $\displaystyle \frac12\left(\frac{2000}{n}-n+1\right)$ is a natural number.

    You'll find that this is satisfied for:
    $\displaystyle n=1, m = 1000$

    $\displaystyle n=5,m=198$

    $\displaystyle n=16,m=55$

    $\displaystyle n=25,m=28$
    I'll leave you to write down the resulting sequences of numbers.

    Grandad
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