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Math Help - Arthimetic progression help

  1. #1
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    Arthimetic progression help

    7. The 17th term of an AP is 22, and the sum of the first 17 terms is 102. Find
    the 1st term and the common difference.

    ANSwer: a1=10;d=2

    can you please help me understand the working
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  2. #2
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    Re: Arthimetic progression help

    Quote Originally Posted by Benja303 View Post
    7. The 17th term of an AP is 22, and the sum of the first 17 terms is 102. Find
    the 1st term and the common difference.
    ANSwer: a1=10;d=2

    Every term of the AP is a_n=a_1+(n-1)d.

    So a_{17}=a_1+16d=22. and \sum\limits_{k = 1}^{17} {{a_k}}  = \sum\limits_{k = 1}^{17} {\left( {{a_1} + (k - 1)d} \right)}  = 17{a_1} + \frac{{16 \cdot 17}}{2}d = ?

    Now it is your turn.
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    MHF Contributor MarkFL's Avatar
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    Re: Arthimetic progression help

    The nth term is:

    a_n=a_1+(n-1)d

    The sum of the first n terms is:

    S_n=\frac{n(a_1+a_n)}{2}

    Use the second equation to find a_1, then use the first to find d. What do you find?
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    Re: Arthimetic progression help

    Ohhh thanks mark. I was doing it was N/2 X the rest. I see . The 2 divides the entire thing. cool. Nice whip by the way.
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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: Arthimetic progression help

    Quote Originally Posted by Benja303 View Post
    Ohhh thanks mark. I was doing it was N/2 X the rest. I see . The 2 divides the entire thing. cool. Nice whip by the way.
    They are actually the same:

    \frac{n(a_1+a_n)}{2}=\frac{n}{2}(a_1+a_n)

    Did you find that a_1 is different than the value you cited in your fist post?
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