2007 UNCO Math Contest II Problems/Problem 1

Problem

Express the following sum as a whole number:

$1+ 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 +10 +11-12 +\cdots + 2005 + 2006 - 2007.$

Solution

We see that the 2nd and the 3rd terms make $-1$ , the 5th and 6th terms make $-1$ , and so on. Therefore, we are left with: $1-1+4-1+7-1+10-1+\dots+2005-1.$ We have two sequences: $1+4+7+10+\dots+2005$ and $-1-1-1-1-\dots-1.$ The first sequence sums to $\dfrac{2006}{2}\cdot 669 = 671007.$ The second sequence includes $669$ terms, so it equals: $669(-1)=-669.$ Summing these two, we have: $671007-669 = \boxed{670338}.$

2007 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions

Page

Toolbox

Search

2007 UNCO Math Contest II Problems/Problem 1

Problem

Solution

See Also