+0  
 
+1
724
3
avatar+10 

What is the value of 
\(1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018? \)

NOTE: This comes from the AMC8 2018 test, but unfortunately, I got really confused about trying to solve it... I want a nudge...

 

EDIT: Ok, I know you have to do something with pairs, just don't know what to do...

 Jun 28, 2019
edited by Questionasker  Jun 28, 2019
 #1
avatar+8963 
+5

=   1 + 3 + 5 + . . . + 2017 + 2019 - 2 - 4 - 6 - . . . - 2016 - 2018

 

(1 - 2) + (3 - 4) + (5 - 6) + . . . + (2015 - 2016) + (2017 - 2018) + 2019

 

=  (1 - 2) + (3 - 4) + (5 - 6) + . . . + (2015 - 2016) + (2017 - 2018) + 2019

 

=  -1   +   -1   +   -1   + . . . +   -1   +   -1   +   2019

 

Now the question is how many  -1 's  are there?

 

the number of  -1 's   =   2018 / 2   =   1009

 

So the sum in question   =   1009( -1 )  +  2019   =   1010

 Jun 28, 2019
 #2
avatar+10 
+1

thx hetictar

Questionasker  Jun 28, 2019
 #3
avatar
+1

Formula: Sum of ODD consecutive numbers =n^2, whre n =number of terms.

Number of terms =[2019 - 1] / 2 + 1 =1010 terms.

n^2 =1010^2 =1,020,100 - sum of consecutine ODD numbers. 

 

Formula: Sum of consecutive EVEN numbers =n x (n + 1), where n = numbers of terms.

Number of terms =[2018 - 2] / 2 + 1 =1009 terms.

n*(n +1) =1009 * 1010 =1,019,090 - sum of consecutive EVEN numbers

 

Difference =sum of ODD numbers - sum of EVEN numbers =1,020,100 -1,019,090 =1010

 Jun 28, 2019

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