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Find all possible ordered triads $$(a, b, c)$$ such that $$(n + 3)^2 = a(n + 2)^2 + b(n + 1)^2 + cn^2$$ for every positive integer $$n$$.
Correct Answer: 0
PLACEHOLDER ANSWER - the answer is an ordered triad. Since the identity holds for infinitely many $$n$$, coefficients must match, giving $$a + b + c = 1$$, $$4a + 2b = 6$$ and $$4a + b = 9$$. Solving, $$b = -3$$, $$a = 3$$ and $$c = 1$$. The only triad is $$(3, -3, 1)$$.
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