Question 12
For all permissible natural numbers $$n$$, the number $$\frac{9n^2-64}{n-1-\frac{1}{1-\frac{n}{n+4}}}$$ is
Solution
We have $$1-\frac{n}{n+4}=\frac{4}{n+4}$$, so the denominator becomes $$n-1-\frac{n+4}{4}=\frac{3n-8}{4}$$. Also, $$9n^2-64=(3n-8)(3n+8)$$. The expression therefore simplifies to $$4(3n+8)$$, which is a natural number divisible by $$4$$ for every permissible natural number $$n$$.
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