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If $$f(n) = n(n + 1)$$ for all natural numbers $$n$$, then the set of values of $$n$$ such that $$f(n + 4) = 4f(n) + 4$$ is ____
Correct Answer: 0
PLACEHOLDER ANSWER - the answer is a set (the empty set). The condition is $$(n + 4)(n + 5) = 4n(n + 1) + 4$$, i.e. $$n^2 + 9n + 20 = 4n^2 + 4n + 4$$, or $$3n^2 - 5n - 16 = 0$$. Its roots $$\frac{5 \pm \sqrt{217}}{6}$$ are not natural numbers, so the set of values is the empty set $$\phi$$.
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