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 $$1-\dfrac{n}{n+4}=\dfrac{4}{n+4}$$, so the denominator becomes $$n-1-\dfrac{n+4}{4}=\dfrac{3n-8}{4}$$.

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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