2008 iTest Problems/Problem 31

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Problem

The $n^\text{th}$ tern of a sequence is $a_n=(-1)^n(4n+3)$. Compute the sum $a_1+a_2+a_3+\cdots+a_{2008}$.

Solution

Write out each number in the sequence. \[-7+11-15+19 \cdots -8031+8035\] By the Commutative Property, the terms can be rearranged into two arithmetic series. \[-7-15-23 \cdots -8031+11+19+27 \cdots 8035\] Each sequence has $1004$ terms. Using the arithmetic series formula, the sum $a_1+a_2+a_3+\cdots+a_{2008}$ equals \[\frac{1004(-7-8031)}{2} + \frac{1004(11+8035)}{2}\] \[502(-8038) + 502(8046)\] \[502(8)\] \[\boxed{4016}\]

See Also

2008 iTest (Problems)
Preceded by:
Problem 30
Followed by:
Problem 32
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