Hkg TST

Algebra Difficulty 7.3 National Olympiad, round 2 Prove it Hong Kong

The general term of a sequence $\{a_n\}$ is given by $a_{n+2} = a_{n+1} - a_n$ . If the sum of the first 1997 terms is 1879 and the sum of the first 1879 terms is 1997, what is the sum of the first 2000 terms?

Solution

The answer is 5873.
Let $a_1 = x$ and $a_2 = y$ . Then the sequence is
$x, y, y-x, -x, -y, x-y, x, y, \dots$
Note that the sequence repeats every 6 terms. Also, the sum of every 6 consecutive terms is 0. Thus, the sum of the first 1997 = $6 \times 332 + 5$ terms is $x + y + (y-x) + (-x) + (-y) = y-x$ , and the sum of the first 1879 = $6 \times 313 + 1$ terms is $x$ . Given that $y-x = 1879$ and $x = 1997$ , we have
$y = (y - x) + x = 1879 + 1997 = 3876.$
Therefore, the sum of the first 2000 = $6 \times 333 + 2$ terms is
$x + y = 1997 + 3876 = 5873.$

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