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: \[671007-669 = \boxed{670338}.\]

See Also

2007 UNCO Math Contest II (ProblemsAnswer KeyResources)
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