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Math Help - Arithmetic sequence

  1. #1
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    Arithmetic sequence

    Pls someone help me:

    {Xn },n = 0,1,2,3, is an arithmetic sequence such that the 7th term minus the 4th term is 12 and the sum of the 7th term and 4th terms is -28.Obtain the closed form of the sequence and write down the 10th and 20th term.

    Tks
    Last edited by watcher; April 5th 2009 at 04:25 AM.
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  2. #2
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    Hello, watcher!

    There is an arithmetic sequence such that the 7th term minus the 4th term is 12
    and the sum of the 7th term and 4th terms is -28.
    Obtain the closed form of the sequence and write down the 10th and 20th terms.
    You're expected to know this formula . . .

    The n^{th} term is: . a_n \;=\;a + (n-1)d
    . . where a is the first term, and d is the common difference.


    The 4^{th} term is: . a_4 \:=\:a + 3d
    The 7^{th} term is: . a_7 \;=\;a + 6d


    Their difference is 12: . (a+6d)-(a+3d) \:=\:12 \quad\Rightarrow\quad 3d \:=\:12 \quad\Rightarrow\quad\boxed{d = 4}\;\;{\color{red}[1]}

    Their sum is -28: . (a+6d)+(a+3d)\:=\:\text{-}28 \quad\Rightarrow\quad 2a+9d \:=\:\text{-}28\;\;{\color{red}[2]}
    . . Substitute [1] into [2]: . 2a + 9(4) \:=\:\text{-}28 \quad\Rightarrow\quad\boxed{a \:=\:\text{-}32}


    Hence, the n^{th} term is: . a_n \;=\;-32 + (n-1)4 \quad\Rightarrow\quad {\color{blue}a_n \:=\:4n-36}


    The 10^{th} term is: . a_{10} \;=\;4(10)-36 \quad\Rightarrow\quad{\color{blue}a_{10}\;=\;4}

    The 20^{th} term is: . a_{20} \;=\;4(20)-36 \quad\Rightarrow\quad{\color{blue}a_{20}\;=\;44}

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  3. #3
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    Reply to Soroban

    Thanks a lot. But I have a question though and am confused and curious to know (am not so good in math).

    Since n starts with zero, why was the formula of Xn = a + (n-1)d used in this problem - and not the closed form of Xn = a + nd (n = 0, 1, 2, ...)?

    Thanks again for your help.

    Quote Originally Posted by Soroban View Post
    Hello, watcher!

    You're expected to know this formula . . .

    The n^{th} term is: . a_n \;=\;a + (n-1)d
    . . where a is the first term, and d is the common difference.


    The 4^{th} term is: . a_4 \:=\:a + 3d
    The 7^{th} term is: . a_7 \;=\;a + 6d


    Their difference is 12: . (a+6d)-(a+3d) \:=\:12 \quad\Rightarrow\quad 3d \:=\:12 \quad\Rightarrow\quad\boxed{d = 4}\;\;{\color{red}[1]}

    Their sum is -28: . (a+6d)+(a+3d)\:=\:\text{-}28 \quad\Rightarrow\quad 2a+9d \:=\:\text{-}28\;\;{\color{red}[2]}
    . . Substitute [1] into [2]: . 2a + 9(4) \:=\:\text{-}28 \quad\Rightarrow\quad\boxed{a \:=\:\text{-}32}


    Hence, the n^{th} term is: . a_n \;=\;-32 + (n-1)4 \quad\Rightarrow\quad {\color{blue}a_n \:=\:4n-36}


    The 10^{th} term is: . a_{10} \;=\;4(10)-36 \quad\Rightarrow\quad{\color{blue}a_{10}\;=\;4}

    The 20^{th} term is: . a_{20} \;=\;4(20)-36 \quad\Rightarrow\quad{\color{blue}a_{20}\;=\;44}
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