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

  • Context: MHB 
  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Sequence Value
Click For Summary
SUMMARY

The sequence presented is defined as follows: $0, 1, -1, 2, 3, -2, 4, 5, 6, -3, 7, 8, 9, 10, -4, 11, 12, 13, 14, 15, 16, -5, ...$. The task is to find the value of $a_{2008}$. The sequence alternates between positive integers and negative integers, with the negative integers decreasing by one after every set of positive integers. The pattern indicates that the negative integers appear after every complete set of consecutive positive integers.

PREREQUISITES
  • Understanding of mathematical sequences and series
  • Familiarity with indexing in sequences
  • Basic knowledge of arithmetic patterns
  • Ability to identify and extend numerical patterns
NEXT STEPS
  • Analyze the pattern of negative integers in the sequence
  • Calculate the position of negative integers relative to positive integers
  • Explore similar sequences to identify common patterns
  • Learn about mathematical induction to prove sequence properties
USEFUL FOR

Mathematicians, students studying sequences, educators teaching arithmetic patterns, and anyone interested in numerical analysis.

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]
 

Similar threads

MHB Evaluate ⌊ 1/a_1+1/a_2+....+1/a_{2008} ⌋
  • · Replies 2 ·
Replies
2
Views
2K
MHB Sequence Challenge: Find $a_{61}+a_{63}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB (Seemingly) Random Sequences
  • · Replies 7 ·
Replies
7
Views
2K
High School Golden Ratio in Collatz-like sequences
  • · Replies 1 ·
Replies
1
Views
2K
MHB Number patterns and sequences - Tn Term
  • · Replies 1 ·
Replies
1
Views
2K
MHB Find the largest value in a sequence
  • · Replies 4 ·
Replies
4
Views
2K
MHB Find the 79th term in the sequence
  • · Replies 2 ·
Replies
2
Views
3K
MHB Geometric Sequence find the 23rd term.
  • · Replies 3 ·
Replies
3
Views
2K
MHB Solving a Challenging Number Sequence Problem
  • · Replies 1 ·
Replies
1
Views
1K
MHB What is the value of (a+b)^3+(b+c)^3+(c+a)^3 for the roots of 8x^3+1001x+2008=0?
  • · Replies 2 ·
Replies
2
Views
1K