MHB Find the Value of $a_{2008}$ in the Sequence

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The sequence presented alternates between positive integers and negative integers, with negative integers appearing at regular intervals. The positive integers increase sequentially, while the negative integers decrease by one after every set of positive integers. To find $a_{2008}$, the pattern must be analyzed to determine the position of the negative integers and the count of positive integers leading up to that index. The sequence continues with this established pattern, allowing for the calculation of $a_{2008}$. The final value of $a_{2008}$ can be derived by following the established rules of the sequence.
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A sequence as follows :
$0,1,-1,2,3,-2,4,5,6,-3,7,8,9,10,-4,11,12,13,14,15,16,-5,--------$
please find :$a_{2008}=?$
 
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Albert said:
A sequence as follows :
$0,1,-1,2,3,-2,4,5,6,-3,7,8,9,10,-4,11,12,13,14,15,16,-5,--------$
please find :$a_{2008}=?$

are you sure that 16 is there
 
kaliprasad said:
are you sure that 16 is there
I am sure that 16 is there .
if 16 is not there , then this question will be much easier
 
Albert said:
A sequence as follows :
$0,1,-1,2,3,-2,4,5,6,-3,7,8,9,10,-4,11,12,13,14,15,16,-5,---------$
please find :$a_{2008}=?$
hint:
we have 1 number after 0
2 numbers after -1(2-3)
3 numbers after -2(4-6)
4 numbers after -3(7-10)
6 numbers after -4(11-16)
and it should have 9 numbers after -5(17-25)
can you figure out the regular pattern?
more hint:
$a_n+a_{n+2}=a_{n+3} \forall n\geq 1$
here we set $a_1=1,a_2=2,a_3=3, a_4=4,a_5=6...$
 
Last edited:
Albert said:
A sequence as follows :
$0,1,-1,2,3,-2,4,5,6,-3,7,8,9,10,-4,11,12,13,14,15,16,-5,--------$
please find :$a_{2008}=?$
[TABLE="width: 270"]
[TR]
[TD="width: 72, bgcolor: transparent"][FONT=&#26032]　
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[TD="width: 72, bgcolor: transparent"][FONT=&#26032]　
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[TD="width: 72, bgcolor: transparent"][FONT=&#26032]　
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[TD="width: 72, bgcolor: transparent"][FONT=&#26032]n
[/TD]
[TD="width: 72, bgcolor: transparent"][FONT=&#26032]an
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]0
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]2
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-1
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]2-3
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]2
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]5
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]3
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-2
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]4-6
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]3
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]9
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]6
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-3
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]7-10
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]4
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]14
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]10
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-4
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]11-16
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]6
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]21
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]16
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-5
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]17-25
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]9
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]31
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]25
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-6
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]26-38
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]13
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]45
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]38
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-7
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]39-57
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]19
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]65
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]57
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-8
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]58-85
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]28
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]94
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]85
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-9
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]86-126
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]41
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]136
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]126
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-10
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]127-186
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]60
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]197
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]186
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-11
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]187-274
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]88
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]286
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]274
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-12
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]275-403
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]129
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]416
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]403
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[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-13
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]404-592
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[TD="bgcolor: transparent"][FONT=&#26032]189
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]606
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]592
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[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-14
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[TD="bgcolor: transparent"][FONT=&#26032]593-869
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[TD="bgcolor: transparent"][FONT=&#26032]277
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[TD="bgcolor: transparent"][FONT=&#26032]884
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[TD="bgcolor: transparent"][FONT=&#26032]869
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[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-15
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[TD="bgcolor: transparent"][FONT=&#26032]870-1275
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[TD="bgcolor: transparent"][FONT=&#26032]406
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[TD="bgcolor: transparent"][FONT=&#26032]1291
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[TD="bgcolor: transparent"][FONT=&#26032]1275
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[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-16
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[TD="bgcolor: transparent"][FONT=&#26032]1276-1870
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]595
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1887
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1870
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-17
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1871-1990
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]120
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]2008
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1990
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