Question 1209331: Hi
Find the value of 1-2+3-4+5-6+.....+2007-2008+2009-2010+2011.
Found 3 solutions by amarjeeth123, ikleyn, math_tutor2020:
Answer by amarjeeth123(570)   (Show Source):
You can put this solution on YOUR website! The given series is 1-2+3-4+5-6+.....+2007-2008+2009-2010+2011
The given series is equivalent to (1+3+5+........+2011)-(2+4+6+........+2010)
We can use Google Sheets and Excel to solve the given problem.
I have attached a snippet of the Excel sheet.
Answer=1006
Answer by ikleyn(52852)  (Show Source):
You can put this solution on YOUR website! .
This problem admits simple mathematical mental solution.
Group the terms in pairs
1-2+3-4+5-6+.....+2007-2008+2009-2010+2011 = (1-2)+(3-4)+(5-6)+.....+(2007-2008)+(2009-2010)+2011.
You have 2010/2 = 1005 pairs and the last unpaired term 2011.
Each pair produces -1, so the total expression is -1005+2011, and you can calculate it mentally.
ANSWER. 1006.
Solved (mentally)
This problem is specially designed to be solved in this way.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Answer: 1006
Explanation
Tutor ikleyn has shown an efficient pathway. Perhaps the most efficient route.
I'll show an alternative method.
Let's place the terms of 1-2+3-4+5-6+...+2007-2008+2009-2010+2011 into a table
| 1 | -2 | 3 | -4 | 5 | -6 | ... | -2006 | 2007 | -2008 | 2009 | -2010 | 2011 |
Then we'll make a copy of this row and reverse it to place under the current terms
| 1 | -2 | 3 | -4 | 5 | -6 | ... | -2006 | 2007 | -2008 | 2009 | -2010 | 2011 | | 2011 | -2010 | 2009 | -2008 | 2007 | -2006 | ... | -6 | 5 | -4 | 3 | -2 | 1 |
Add straight down to see each column adds to either 2012 or -2012.
More specifically the odd values add to 2012 while the even values add to -2012.
In the set {1,2,3,...,2010,2011} there are 2010/2 = 1005 even numbers.
Those even numbers are 2,4,6,...,2010.
This means that we have 1005 instances of the sum -2012 show up.
The other 2011-1005 = 1006 sums are 2012
We then have
1006*2012 + 1005*(-2012)
= 2012*(1006-1005)
= 2012*(1)
= 2012
This is not the final answer.
It would be nice if 1-2+3-4+...+2009-2010+2011 did evaluate to 2012.
However, when I made that 2nd table, where the bottom row is the reverse of the top row, I introduced a second copy of the sum. Thereby the result of adding everything in that 2nd table would be 2*S, where S = 1-2+3-4+...+2009-2010+2011
So we have to divide by 2 to correct this error.
2012/2 = 1006 is the final answer.
In other words, S = 1-2+3-4+...+2009-2010+2011 represents combining terms along the top row
S = 2011-2010+2009-...-4+3-2+1 also happens when we combine terms along the bottom row.
Add those equations straight down to arrive at 2S = 2012 which leads to S = 1006
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