Question

Find the explicit formula that produces the given sequence. 3/2, 3/4, 3/8, 3/16,...

Answer 1

$\frac{3}{2};\ \frac{3}{4};\ \frac{3}{8};\ \frac{3}{16};\ ...\\\\a_1=\frac{3}{2}\\\\a_2=\frac{3}{4}=\frac{3}{2}\cdot\frac{1}{2}\\\\a_3=\frac{3}{8}=\frac{3}{4}\cdot\frac{1}{2}=\frac{3}{2}\cdot\left(\frac{1}{2}\right)^2\\\\a_4=\frac{3}{16}=\frac{3}{8}\cdot\frac{1}{2}=\frac{3}{2}\cdot\left(\frac{1}{2}\right)^3\\\vdots$
$a_n=\frac{3}{2}\cdot\left(\frac{1}{2}\right)^{n-1}=\frac{3}{2}\cdot\left(\frac{1}{2}\right)^{-1}\cdot\left(\frac{1}{2}\right)^n=\frac{3}{2}\cdot2\cdot\left(\frac{1}{2}\right)^n=3\cdot\left(\frac{1}{2}\right)^n$