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Math Help - Partial Sum/Sequence questions...

  1. #1
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    Partial Sum/Sequence questions...

    Obviously I'm making some sort of systematic error...any help is greatly appreciated!

    15.) The number of cars sold weekly by a new auto dealership grows according to a linear growth model. The first week the dealership sold two cars P_0=2 and the second week the dealer ship sold six cars P_1=6

    In the first 50 weeks, how many cars did the dealership sell?

    My work:

    S_n=\frac{(P_0+P_{n-1})(n)}{2}

    P_{n-1}=P_{49-1}=P_{48}

    P_{48}=P_0+nd=2+48(4)=194

    S_{49}=\frac{(2+194)(49)}{2}

    S_{49}=4802

    The correct answer is 5000...



    17.) 5+8+11+14+...+299+302=? (100 terms)

    My work:

    S_{99}=\frac{(5+299)(99)}{2}=15,048

    The correct answer is 15,350



    19.) 15+11+7+3+...

    Find the sum of the first 100 terms

    My work:

    P_{98}=15+(-4)(98)=-377

    S_{99}=\frac{(P_0+P_{n-1})(n)}{2}=\frac{(15+(-377))(99)}{2}=-17,919

    Correct Answer: -18,300



    Like I said, I'm pretty sure it's a systematic error I'm making...I've tried it lots more way besides this, but I think this is the "most" correct way I've tried....any help will be wonderful. Thanks!!
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elizsimca View Post



    17.) 5+8+11+14+...+299+302=? (100 terms)



    The correct answer is 15,350
    \begin{aligned}\underbrace{5+8+11+\cdots}_{100\tex  t{ terms}}&=\sum_{n=0}^{99}\left\{5+3n\right\}\\<br />
&=5(99+1)+3\cdot\frac{99\cdot{100}}{2}\\<br />
&=15350\end{aligned}

    Using \sum_{n=0}^{N}c=c(N+1)

    and \sum_{n=0}^{N}n=\frac{N(N+1)}{2}

    And
    19.) 15+11+7+4+\cdots=?

    100 terms

    Find the sum of the first 100 terms
    \begin{aligned}\underbrace{15+11+7+4+\cdots}_{100\  text{ terms}}&=\sum_{n=0}^{99}\left\{15-4n\right\}\\<br />
&=15(99+1)-4\cdot\frac{99\cdot100}{2}\\<br />
&=-18300\end{aligned}
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  3. #3
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    Quote Originally Posted by Mathstud28 View Post
    \begin{aligned}\underbrace{5+8+11+\cdots}_{100\tex  t{ terms}}&=\sum_{n=0}^{99}\left\{5+3n\right\}\\<br />
&=5(99+1)+3\cdot\frac{99\cdot{100}}{2}\\<br />
&=15350\end{aligned}

    Using \sum_{n=0}^{N}c=c(N+1)

    and \sum_{n=0}^{N}n=\frac{N(N+1)}{2}

    And

    \underbrace{15+11+7+4+\cdots}_{100\text{ terms}}&=\sum_{n=0}^{99}\left\{15-4n\right\}\\<br />
&=15(99+1)-4\cdot\frac{99\cdot100}{2}\\<br />
&=-18300\end{aligned}
    Mathstud,

    So am I using an incorrect formula? Is the formula I tried to use on number 17 completely incorrect? That's the formula my professor gave us and I'm wondering if it was typed incorrectly because I'm getting all my partial arithmetic sums incorrect when I use that formula...when I do it your way, I get the correct answer....any thoughts?

    Thanks
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elizsimca View Post
    Mathstud,

    So am I using an incorrect formula? Is the formula I tried to use on number 17 completely incorrect? That's the formula my professor gave us and I'm wondering if it was typed incorrectly because I'm getting all my partial arithmetic sums incorrect when I use that formula...when I do it your way, I get the correct answer....any thoughts?

    Thanks
    No! You are in fact using the right formula ...but unfortunately you are putting in the wrong numbers

    S_{100}=\frac{\left(5+{\color{red}302}\right)\cdot  {\color{red}100}}{2}=15350

    Dont worry those upper indexes always get people
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  5. #5
    Senior Member JaneBennet's Avatar
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    Quote Originally Posted by elizsimca View Post
    15.) The number of cars sold weekly by a new auto dealership grows according to a linear growth model. The first week the dealership sold two cars P_0=2 and the second week the dealer ship sold six cars P_1=6

    In the first 50 weeks, how many cars did the dealership sell?

    My work:

    S_n=\frac{(P_0+P_{n-1})(n)}{2}

    P_{n-1}=P_{49-1}=P_{48}

    P_{48}=P_0+nd=2+48(4)=194

    S_{49}=\frac{(2+194)(49)}{2}

    S_{49}=4802

    The correct answer is 5000...
    You are taking n=50 so you should be working out S_{50}=\frac{(P_0+P_{49})(50)}2.
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  6. #6
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    Quote Originally Posted by Mathstud28 View Post
    No! You are in fact using the right formula ...but unfortunately you are putting in the wrong numbers

    S_{99}=\frac{\left(5+{\color{red}302}\right)\cdot{  \color{red}100}}{2}=15350

    Dont worry those upper indexes always get people

    But if P_0= Week 1...then wouldn't the 50th week be given by P_{49}?

    Sorry, I'm having a dumb math moment....lol
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elizsimca View Post
    But if P_0= Week 1...then wouldn't the 50th week be given by P_{49}?

    Sorry, I'm having a dumb math moment....lol
    But how many terms are there?
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  8. #8
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    Quote Originally Posted by Mathstud28 View Post
    But how many terms are there?
    touche'
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