10, 8, 6, 4, 2, 0, -2, -4
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It looks like the Fibonacci sequence grouped in two terms and up. The next should be 8132134.
You cant solve the next term (next number) in this sequence. You need more terms, because this is either a "quadratic sequence", or a "linear and quadratic sequence", and you need more terms than this to solve a "linear and quadratic sequence" and for this particular "quadratic sequence" you would need more terms to solve nth term, which would solve what the next number is. If this is homework, check with your teacher if he wrote the wrong sum.
The next number after 17799 is 17800. This follows the conventional counting sequence where each number increases by one.
Fibonacci sequence