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Math Help - PLEASE HELP - Arithmetic Sequences

  1. #1
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    PLEASE HELP - Arithmetic Sequences

    Key, just in case there are varying letters used
    U - unknown number
    n - number you're looking for
    d - common difference

    Find the general term for Un for an arithmetic sequence given that:
    U4=37 and U10 =67

    I know how to do it up to here, but I don't know what to do next:
    U10-U4=6d
    --> U10-U4=67-37=30
    6d=30
    d=5

    So I have the common difference but I don't know what to do with it now, even if I just insert it into the equation, I don't have the other values...

    Un=U1+(n-1)d
    Un=U1+(n-1)(5)
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  2. #2
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    Hello, juliak!

    Find the general term U_n for an arithmetic sequence given that:
    . . . U_4=37\:\text{ and }\:U_{10} =67

    We're expected to know that the n^{th} term is: . U_n \:=\:U_1 + (n-1)d

    . . where: . \begin{Bmatrix}U_1 &=& \text{First term} \\ n &=& \text{no. of terms} \\ d &=& \text{common diff.} \end{Bmatrix}


    We're told that: . \begin{array}{cccc}U_4 = 37  & \Rightarrow  & U_1 + 3d \:=\:37 & {\color{blue}[1]} \\<br />
U_{10} = 67 & \Rightarrow &  U_1 + 9d \:=\:67 & {\color{blue}[2]}\end{array}


    Now solve the system of equations . . .

    . . Subtract [1] from [2]: . 6d \:=\:30 \quad\Rightarrow\quad \boxed{d \:=\:5}

    . . Substitute into [1]: . U_1 + 3(5) \:=\:37 \quad\Rightarrow\quad \boxed{U_1 \:=\:22}


    The sequence is: . 22,\:27,\:32,\:37,\:42,\:27,\:\hdots

    The general term is: . U_n \:=\:22 + (n-1)5 \quad\Rightarrow\quad\boxed{ {\color{blue}U_n \:=\:5n + 17}}

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