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Math Help - arithmetic series

  1. #1
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    arithmetic series

    Task: explain step-by-step how to find the sum of the number 1 to 2000 that are not divisible by 3 or 7. Be sure your plan includes steps to check your work.

    Because you would already know how, Iíll give you the quick version of what I did. (not explaining exactly what each value is)
    The only formula that I used was Sn= [n(a1+an)]/2

    Step one. Find sum of numbers 1 to 2000 using formula. (Equals 2001000)
    Step two, find sum of numbers 1-2000 that are divisible by 3 (Equals 666333)
    Step three find sum of numbers 1-2000 that are divisible by 7 (Equals 285,285)
    Step four Subtract answers 2 and 3 from 1.
    Step five, find sum of numbers 1-2000 that are divisible by 21 and add it back so that you arenít subtracting the same numbers twice.
    To checkÖ
    This is where I draw a blank. I have No idea. Please help.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Cheeta921 View Post
    Task: explain step-by-step how to find the sum of the number 1 to 2000 that are not divisible by 3 or 7. Be sure your plan includes steps to check your work.

    Because you would already know how, Iíll give you the quick version of what I did. (not explaining exactly what each value is)
    The only formula that I used was Sn= [n(a1+an)]/2

    Step one. Find sum of numbers 1 to 2000 using formula. (Equals 2001000)
    Step two, find sum of numbers 1-2000 that are divisible by 3 (Equals 666333)
    Step three find sum of numbers 1-2000 that are divisible by 7 (Equals 285,285)
    Step four Subtract answers 2 and 3 from 1.
    Step five, find sum of numbers 1-2000 that are divisible by 21 and add it back so that you arenít subtracting the same numbers twice.
    To checkÖ
    This is where I draw a blank. I have No idea. Please help.
    add to it the numbers divisible by 4,5,6,7,8,9 this will provide the sum of all numbers if you did it correctly
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  3. #3
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    Krizalid's Avatar
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    Quote Originally Posted by Cheeta921 View Post
    Step one. Find sum of numbers 1 to 2000 using formula. (Equals 2001000)
    Step two, find sum of numbers 1-2000 that are divisible by 3 (Equals 666333)
    Step three find sum of numbers 1-2000 that are divisible by 7 (Equals 285,285)
    Step four Subtract answers 2 and 3 from 1.
    Step five, find sum of numbers 1-2000 that are divisible by 21 and add it back so that you aren’t subtracting the same numbers twice.
    To check…
    This is where I draw a blank. I have No idea. Please help.
    You're expected to know that \sum_{k\,=\,1}^n k=\frac{n(n+1)}2, this will do those stuff.
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  4. #4
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    Thank you, Krizalid, But i don't know what you mean by that formula. Could you please explain?
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  5. #5
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    Quote Originally Posted by Cheeta921 View Post
    Step one. Find sum of numbers 1 to 2000 using formula. (Equals 2001000)
    Suppose you want to find 1+2+3+\cdots+n, eventually for n=2000 we want to find 1+2+3+\cdots+2000. Do you understand what \sum means? The formula I gave you solves those problems.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Cheeta921 View Post
    Thank you, Krizalid, But i don't know what you mean by that formula. Could you please explain?
    It is the same formula you have in your passage... 1+2+3+4+5+6+...+(n-1)+n=\sum_{k=1}^{n}k=\frac{n(n+1)}{2}
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  7. #7
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    Right, My book uses i instead of K... I used the other formula since it means the same thing, could i use this one as my check? Thank you for being patient with me.
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