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Algebra Difficulty 3.8 AMC 10/12 Find the answer Slovenia

The sum of every three consecutive terms of the sequence $a_1, a_2, a_3, a_4, \dots$ equals $2016$ . It also holds that $a_{667} = 667$ and $a_{1004} = 1004$ . What is the value of $a_{2016}$ ?

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Solution

Since the sum of any three consecutive terms is the same we have $a_{n+3} = a_n$ for all $n$ . This implies that $a_{667} = a_{667+3 \cdot 449} = a_{2014}$ and $a_{1004} = a_{1004+3 \cdot 337} = a_{2015}$ . From $a_{2014} + a_{2015} + a_{2016} = 2016$ we conclude that $a_{2016} = 2016 - 667 - 1004 = 345$ .

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