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...

Questionasker Jun 28, 2019

#1**+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

hectictar Jun 28, 2019

#3**+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**

Guest Jun 28, 2019