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Thread: Problem on arithmetic sequences

  1. #1
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    Problem on arithmetic sequences

    Let $\{a_n\}$ and $\{A_n\}$ be two arithmetic sequences, and $s_n$ and $S_n$ sums of ther first $n$ terms.
    $$If\ \ \frac{s_n}{S_n}=\frac{7n+1}{4n+27},\ calculate\ \ \frac{a_{11}}{A_{11}}$$.
    Last edited by ns1954; Jan 26th 2019 at 01:06 AM.
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  2. #2
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    Re: Problem on arithmetic sequences

    What have you tried?
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  3. #3
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    Re: Problem on arithmetic sequences

    $$\frac{a_{11}}{A_{11}}=\frac{a_1+10d}{A_1+10D}$$

    I supose problem must transform in system of 3 equation with 3 unknown, $a_1$, $d$, and $D$, ($d,D$ coresponding differences)
    Last edited by ns1954; Jan 26th 2019 at 07:49 AM.
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  4. #4
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    Re: Problem on arithmetic sequences

    Hint:
    nth term:
    a + d(n-1)
    A + D(n-1)

    So 11th term:
    a + 10d
    A + 10D

    Also, using sum of 1st nth terms:
    (2a + dn - d)/(2A + Dn - D) = (7n + 1)/(4n + 27)
    Now substitute n = 11
    Last edited by DenisB; Jan 26th 2019 at 08:20 AM.
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    Re: Problem on arithmetic sequences

    Here's my approach:

    Since $\displaystyle \frac{s_n}{S_n} =\frac{7n+1}{4n+27}$


    then let $\displaystyle s_n = k(7n+1)$ and $\displaystyle S_n = k(4n+27)$


    Then use the fact that $\displaystyle a_1 = s_1$ and $\displaystyle a_2 =s_2 - s_1$ to find d.

    Do the same for A, S and D.

    Then see what you can do.
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  6. #6
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    Re: Problem on arithmetic sequences

    I think you are wrong. Your k isn't constant, it depends on n.
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    Re: Problem on arithmetic sequences

    Quote Originally Posted by ns1954 View Post
    I think you are wrong. Your k isn't constant, it depends on n.
    Yes, you are correct. Maybe let k =f(n), but I think that will get messy. Back to the drawing board. What else have you tried?
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    Re: Problem on arithmetic sequences

    ns1954,
    Do you actually have the answer for this question?
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  9. #9
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    Re: Problem on arithmetic sequences

    Do we have the whole question? I have far too many variables at the end and not enough conditions to specify them.

    ala DenisB:

    For the series $\displaystyle a_1,~a_1 + d,~a_1 + 2d,~ \text{...}$
    we have $\displaystyle s_k = \dfrac{k}{2} (a_1 + a_k)$

    So for k = 11: $\displaystyle s_{11} = \dfrac{11}{2} ( a_1 + a_{11})$

    Similarly for the A series:
    $\displaystyle S_{11} = \dfrac{11}{2} (A_1 + A_{11})$

    and we have
    $\displaystyle \dfrac{s_{11}}{S_{11}} = \dfrac{a_1 + a_{11}}{A_1 + A_{11}} = \dfrac{7 \cdot 11 + 1}{4 \cdot 11 + 27} = \dfrac{78}{71}$

    Cross mulitplying, grouping, etc. gives
    $\displaystyle \dfrac{a_{11}}{A_{11}} = \dfrac{78}{71} \left ( \dfrac{A_1}{A_1 + 10D} \right ) - \left ( \dfrac{a_1}{A_1 + 10D} \right ) + \dfrac{78}{71}$

    Oddly enough this is independent of d. But still, we have three unknowns here: $\displaystyle a_1,~A_1,~D$. I can't go any further.

    -Dan
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  10. #10
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    Re: Problem on arithmetic sequences

    Condition
    $$ \frac{s_n}{S_n} =\frac{7n+1}{4n+27}$$
    holds for every n. I think it must be used for n=1,2,3,4 to give enaf equations, but system will be homogenous, and this is a way I hate. I hope it exist some more elegant way. I don't have solution, only answer $\frac{a_{11}}{A_{11}}=\frac{4}{3}$.
    Last edited by ns1954; Jan 26th 2019 at 09:42 PM.
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