2014 even numbers
What is the difference between the sum of the first 2014 odd numbers and the sum of the first 2014 even numbers?
Age
11 to 14
Challenge level
Problem
What is the difference between the sum of the first 2014 odd numbers and the sum of the first 2014 even numbers?
This problem is taken from the World Mathematics Championships
Student Solutions
Answer: 2014
Comparing the two sums
Sum of the first 2014 even numbers is $2+4+6+8+10+12+...$
Sum of the first 2014 odd numbers is $1+3+5+7+\ 9\ +11+...$
Difference is $1+1+1+1+\ 1\ +\ 1\ +...$
There are 2014 terms - so the difference is 2014.
Working it out by considering shorter sums
First 2 evens/odds
2 + 4 = 6
1 + 3 = 4
The difference between 6 and 4 is 2.
First 3 evens/odds
2 + 4 + 6 = 12
1 + 3 + 5 = 9
The difference between 12 and 9 is 3.
First 4 evens/odds
2 + 4 + 6 + 8 = 20
1 + 3 + 5 + 7 = 16
The difference between 20 and 16 is 4.
Assuming the pattern continues, the difference between the sum of the first 2014 even numbers and the sum of the first 2014 odd numbers will be 2014.