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Math Help - Sequences equal?

  1. #1
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    Sequences equal?

    There are two sequences:
    a_1,a_2,...; \:\:\:\:\:<br />
b_1,b_2,... and each a_i, b_i element of the sequences is 2 or 3. We know that:
    \frac 1 a_1 + \frac 1 {a_1a_2} + \frac 1 {a_1a_2a_3} + ... = \frac 1 b_1 + \frac 1 {b_1b_2} + \frac 1 {b_1b_2b_3} + ...
    Prove that the two sequences are equals.

    Any help would be appreciated!
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  2. #2
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    Quote Originally Posted by doug View Post
    There are two sequences:
    a_1,a_2,...; \:\:\:\:\:<br />
b_1,b_2,... and each a_i, b_i element of the sequences is 2 or 3. We know that:
    \frac 1 a_1 + \frac 1 {a_1a_2} + \frac 1 {a_1a_2a_3} + ... = \frac 1 b_1 + \frac 1 {b_1b_2} + \frac 1 {b_1b_2b_3} + ...
    Prove that the two sequences are equals.

    Any help would be appreciated!

    By reductio ad absurdum. First, show that it is enough to suppose a_1\neq b_1 (this already is a little nice problem!),

    say a_1=2\,,\,b_1=3.

    Now, show that \displaystyle{\frac{1}{2}+\frac{1}{2a_2}+\ldots\ge  q \frac{3}{4}\,,\,\,and\,\,\frac{1}{3}+\frac{1}{3b_2  }+\ldots\leq \frac{2}{3}} and get a contradiction.

    Tonio
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