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Thread: Radical without calculator

  1. #1
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    Radical without calculator

    Let a = 111 . . . 1, where the digit 1 appears 2018 consecutive times. Let b = 222 . . . 2, where
    in this case the digit 2 appears 1009 consecutive times. Without using a calculator, evaluate
    √(a − b)
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  2. #2
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    Re: Radical without calculator

    Quote Originally Posted by SAMRAL View Post
    Let a = 111 . . . 1, where the digit 1 appears 2018 consecutive times. Let b = 222 . . . 2, where
    in this case the digit 2 appears 1009 consecutive times. Without using a calculator, evaluate
    √(a − b)
    Here is what you are asked to do: $\sqrt {\sum\limits_{k = 0}^{2018} {{{10}^k}} - 2 \cdot \sum\limits_{k = 0}^{1009} {{{10}^k}} }$.

    Now I have no idea what the point of this question may be. Nor can I imagine what it is designed to teach.

    With a calculator see here.

    Without, I think this probability busywork.
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  3. #3
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    Re: Radical without calculator

    That's wrong. The upper indices are incorrect. The actual answer is pretty straightforward, but I haven't really thought about how to get to it.
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  4. #4
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    Re: Radical without calculator

    for all $n$

    $\sum _{k=0}^{2n-1} 10^k-2\sum _{k=0}^{n-1} 10^k=\left(3\sum _{k=0}^{n-1} 10^k\right){}^2$

    to prove this, note that each sum is a geometric series so for example

    $\sum _{k=0}^{2n-1} 10^k=\frac{10^{2n}-1}{9}$

    etc.
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