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

  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Sequence Value
AI Thread Summary
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.
Albert1
Messages
1,221
Reaction score
0
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 :$a_{2008}=?$
 
Mathematics news on Phys.org
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] 
[/TD]
[TD="width: 72, bgcolor: transparent"][FONT=&#26032] 
[/TD]
[TD="width: 72, bgcolor: transparent"][FONT=&#26032] 
[/TD]
[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
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-13
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]404-592
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]189
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]606
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]592
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-14
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]593-869
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]277
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]884
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]869
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-15
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]870-1275
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]406
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1291
[/TD]
[TD="bgcolor: transparent"][FONT=&#26032]1275
[/TD]
[/TR]
[TR]
[TD="bgcolor: transparent"][FONT=&#26032]-16
[/TD]
[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
[/TD]
[/TR]
[/TABLE]
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

Similar threads

Replies
2
Views
2K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
7
Views
2K
Replies
1
Views
2K
Replies
1
Views
2K
Replies
4
Views
2K
Replies
4
Views
2K
Replies
2
Views
3K
Replies
3
Views
2K
Back
Top