JAMB 2000 · UME · Q9

Simplify $\dfrac{3(2^{n + 1}) - 4(2^{n - 1})}{2^{n + 1} - 2^n}$ .

Worked solution (try it first)

Write each power in terms of $2^n$ : $2^{n + 1} = 2 \times 2^n$ and $2^{n - 1} = \frac12 \times 2^n$ .
Top: $3 \times 2 \times 2^n - 4 \times \frac12 \times 2^n = 6 \times 2^n - 2 \times 2^n$
$= 4 \times 2^n$ .
Bottom: $2 \times 2^n - 2^n = 2^n$ .
So the fraction is $\frac{4 \times 2^n}{2^n} = 4$ , option C.