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Math Help - Help with pattern problem

  1. #1
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    Help with pattern problem

    Hi,

    please could someone assist with the following:
    This is the pattern give:
    Pattern number 1 2 3 4
    Number of dots 2 6 12 20

    The question is: Find s formula for the general term

    Their solution goes as follows:
    The second difference is constant therefore Tn is quadratic - I understand this
    therefore Tn = an2+bn+c this i understand
    but 2a=a - I cannot understand the basis of this statement
    Can anyone enlighten me?

    Thanks!
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  2. #2
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    Quote Originally Posted by robyn6680 View Post
    Hi,

    please could someone assist with the following:
    This is the pattern give:
    Pattern number 1 2 3 4
    Number of dots 2 6 12 20

    The question is: Find s formula for the general term

    Their solution goes as follows:
    The second difference is constant therefore Tn is quadratic - I understand this
    therefore Tn = an2+bn+c this i understand
    but 2a=a - I cannot understand the basis of this statement
    Can anyone enlighten me?

    Thanks!
    So the source said 2a = a? Then it's a typo, because this would mean a = 0, which clearly can't be since then we'd have a linear closed form solution.
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  3. #3
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    Number of dots 2 6 12 20: oblong numbers.

    Does this help?
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  4. #4
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    Quote Originally Posted by robyn6680 View Post
    Hi,

    please could someone assist with the following:
    This is the pattern give:
    Pattern number 1 2 3 4
    Number of dots 2 6 12 20

    The question is: Find s formula for the general term
    note the number of dots rewritten as a product ...

    1 \cdot 2, \, 2 \cdot 3, \, 3 \cdot 4, \, 4 \cdot 5, \, ... \,  , \, n(n+1), \, ...
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  5. #5
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    Quote Originally Posted by robyn6680 View Post
    Hi,

    please could someone assist with the following:
    This is the pattern give:
    Pattern number 1 2 3 4
    Number of dots 2 6 12 20

    The question is: Find s formula for the general term

    Their solution goes as follows:
    The second difference is constant therefore Tn is quadratic - I understand this
    therefore Tn = an2+bn+c this i understand
    but 2a=a - I cannot understand the basis of this statement
    Can anyone enlighten me?

    Thanks!
    You say you understand that T_n= an^2+ bn+ c. Since you have 3 numbers to determine, a, b, and c, you need 3 equations. Pick any 3 numbers from the sequence.

    Since T_1= 2, a(1^2)+ b(1)+ c= a+ b+ c= 2.

    since T_2= 6, a(2^2)+ b(2)+ c= 4a+ 2b+ c= 6

    Since T_3= 12, a(3^2)+ b(3)+ c= 9a+ 3b+ c= 12

    Subtract the first equation from the second to get (4a+ 2b+ c)- (a+ b+ c)= 3a+ b= 6- 2= 4 or 3a+ b= 4. c has been eliminated.

    Subtract the first equation from the third to get (8a+ 3b+ c)- (a+ b+ c)= 7a+ 2b= 12- 2= 10 or 7a+ 2b= 10. Again c has been eliminated. Now we have the two equations 3a+ b= 4 and 7a+ 2b= 10.

    Multiply 3a+ b= 4 by 2 to get 6a+ 2b= 8 and subtract that from 7a+ 2b= 10: (7a+ 2b)- (6a+ 2b)= a= 10- 8= 2.

    a= 2, not "2a= a".

    Now that you know a= 2, use either 3a+ b= 4 or 7a+ 2b= 10 to determine b. Then use any of the original 3 equations to determine c.

    As a check, see if those values for a, b, and c give T_4= 20.
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  6. #6
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    Hi HallsofIvy,
    you have a typo there....(9a+3b+c)-(a+b+c)

    We end up with a=1, b=1, c=0 leading to skeeter's formula.
    Last edited by Archie Meade; August 4th 2010 at 06:46 AM.
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