Question 2

The number of natural numbers $$n$$ satisfying the equation $$\frac{8^n - 2^n}{6^n - 3^n} = 2$$ is

Solution

For $$n = 1$$ the left side is $$\frac{8 - 2}{6 - 3} = 2$$, so $$n = 1$$ works. The equation is equivalent to $$8^n - 2^n = 2 \times 6^n - 2 \times 3^n$$, and for $$n = 2$$ the two sides are $$60$$ and $$54$$, with the left side growing faster afterwards because $$8^n$$ dominates. Hence the equality holds for exactly one natural number.

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