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Thread: Sigma notation

  1. #1 Sigma notation 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    hey guys, im trying to prove the Volume of cone using Series and limits.
    i have been doing so good so far, until i reached this point of the solution,,

    =

    i just dont get how that sum notation suddenly transformed in to is it something to do with the arithmetic progression or the geometric?

    i would appreciate any clear approach to my problem. thank you.


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  3. #2 Re: Sigma notation 
    . DrRocket's Avatar
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    Quote Originally Posted by Heinsbergrelatz
    hey guys, im trying to prove the Volume of cone using Series and limits.
    i have been doing so good so far, until i reached this point of the solution,,

    =

    i just dont get how that sum notation suddenly transformed in to is it something to do with the arithmetic progression or the geometric?
    Prove the formula by induction. It is straightforward.


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  4. #3  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    i not m\ familiar with mathematical induction, can you show me the steps on how to induce this particular problem?
    i would appreciate it.
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  5. #4  
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    Quote Originally Posted by Heinsbergrelatz
    i not m\ familiar with mathematical induction, can you show me the steps on how to induce this particular problem?
    i would appreciate it.
    Proof by induction goes as follows. To start you have an assertion A(n) that is to hold for all natural number n. There are 2 steps to a proof by induction.

    1. First you show that A(1) is true.

    2. Then you show that if A(n) is true then it follows that A(n+1) must also be true.

    In this case we assert that



    Now,



    So the assertion is true for n=1

    Now assume that it true for some arbitrary n.













    QED
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  6. #5  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    thank you for your clear solutions, but i get the second part where you add the , but i dont get how on earth did the Sigma k^{2}=became that fractional function.
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  7. #6  
    . DrRocket's Avatar
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    Quote Originally Posted by Heinsbergrelatz
    thank you for your clear solutions, but i get the second part where you add the , but i dont get how on earth did the Sigma k^{2}=became that fractional function.
    I came from the assumption that the assertion is true for some n.

    Get your teacher to explain to you "proof by induction".
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  8. #7  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    Induction is the topic for next year, in the IB diploma mathematics... But i am studying mathematical induction and it doesn't seem a great challenge, but it gets confusing surely.
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  9. #8  
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    Quote Originally Posted by Heinsbergrelatz
    Induction is the topic for next year, in the IB diploma mathematics... But i am studying mathematical induction and it doesn't seem a great challenge, but it gets confusing surely.
    Induction is extremely simple.

    It is based on the axiom that if you have a subset A of the natural numbers such that 1 is in A and if whenver n is in A then n+1 is also in A that A is in fact the entire set of natural numbers.

    I am rather surprised that you have not been exposed to proof by induction before. In my experience it is taught long before one is introduced to calculus.
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  10. #9 Re: Sigma notation 
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    Quote Originally Posted by Heinsbergrelatz
    hey guys, im trying to prove the Volume of cone using Series and limits.
    i have been doing so good so far, until i reached this point of the solution,,

    =

    i just dont get how that sum notation suddenly transformed in to is it something to do with the arithmetic progression or the geometric?

    i would appreciate any clear approach to my problem. thank you.
    Here I introduce a summation formula I found myself. Just consider the sequences below.

    a1
    a1 a2
    ...
    a1 a2 ... an

    The summation of all the values above equals T1=

    a1 a2 ... an
    a2 ... an
    ...
    an

    The summation of all the values above equals T2=

    And implicitly we can see that T1+T2=
    And we get
    =
    U can work out that problem by employing this formula.
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  11. #10  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    Induction is extremely simple.

    It is based on the axiom that if you have a subset A of the natural numbers such that 1 is in A and if whenver n is in A then n+1 is also in A that A is in fact the entire set of natural numbers.

    I am rather surprised that you have not been exposed to proof by induction before. In my experience it is taught long before one is introduced to calculus.
    it is very comprehensible that you notice the syllabus very bizarre. But we are now on the IGCSE year 10~11 course and, normally we dont get introduced to calculus and all those complicated mathematics. It is just standard Maths. But there is an extra topic called "Additional Mathematics" where it lets you study 2years of all the calculus,vectors and relative velocity etc... before you enter the A-level and IB diploma. So this is just like a preparation for the students who will be taking pure mathematics in the future courses\, like me, who will be going for IB maths higher. now this is where you actually properly define and understand concepts of mathematics in a deeper scale before entering University.
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  12. #11  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    i just realized, that;



    =

    and therefore;

    =
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  13. #12  
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    Quote Originally Posted by Heinsbergrelatz
    i just realized, that;



    =

    and therefore;

    =
    There are a couple of problems.

    First with notation. What meant to write is

    =

    Secondly, what you wrote is false. Both equalities are wrong.

    = unless

    The correct expression is



    You might want to prove this for yourself using induction. It is not hard.
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  14. #13  
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    just finished proving them.
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