+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...
+9466 = 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
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
