Simplify $\dfrac{2^{n+4} - 2 \cdot 2^{n}}{2 \cdot 2^{n+3}}$.

Simplify $\dfrac{2^{n+4} - 2 \cdot 2^{n}}{2 \cdot 2^{n+3}}$.
  1. $\dfrac{7}{8}$
  2. $\dfrac{1}{8}$
  3. $\dfrac{7}{16}$
  4. $1$

Solution

Factor $2^{n}$: numerator $= 2^{n}(2^{4} - 2) = 14 \cdot 2^{n}$, denominator $= 2 \cdot 2^{n+3} = 16 \cdot 2^{n}$. Ratio $= \dfrac{14}{16} = \dfrac{7}{8}$.

Asked in: IMO

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