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Thread: Series...

  1. #1
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    Series...

    u = sum of an arithmetic series, 4 terms, common difference = 3

    v = sum of a geometric series, 4 terms, multiplier = 3

    Let it be known that u - v = 162

    For your added pleasure, 1st terms are the same

    Your mission: find the 1st term.
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  2. #2
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    Re: Series...

    Hello, DenisB!

    u = sum of an arithmetic series, 4 terms, common difference = 3

    v = sum of a geometric series, 4 terms, common ratio = 3

    Let it be known that: u - v \:=\: 162

    For your added pleasure, 1st terms are the same

    Your mission: find the 1st term.

    Let a = first term.

    u \;=\;\sum \begin{Bmatrix} a \\ a+3 \\ a+6 \\ a+9 \end{Bmatrix}\quad\Rightarrow\quad u \:=\:4a+18

    v \;=\;\sum \begin{Bmatrix} a \\ 3a \\ 9a \\ 27a \end{Bmatrix} \quad\Rightarrow\quad v \:=\:40a

    u-v \:=\:162 \quad\Rightarrow\quad (4a+18) - 40a \:=\:162

    . . . -36a \;=\;144 \quad\Rightarrow\quad \boxed{a \;=\;-4}
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  3. #3
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    Re: Series...

    Perfecto! Can't fool you, Soroban.

    -4, -1, 2, 5 : 2
    -4,-12,-36,-108 : -160
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