Question

What is the recursive rule for this geometric sequence? 2, 1/2, 1/8,  1/32,  

Answer 1

$a_1=2,\ a_2=\dfrac{1}{2},\ a_3=\dfrac{1}{8};\ a_4=\dfrac{1}{32}\\\\r=\dfrac{a_2}{a_1}=\dfrac{a_3}{a_2}=\dfrac{a_4}{a_3}=...\\\\r=\dfrac{\frac{1}{2}}{2}=\dfrac{\frac{1}{8}}{\frac{1}{2}}=\dfrac{\frac{1}{32}}{\frac{1}{8}}=\dfrac{1}{4}\\\\\boxed{a_1=2;\qquad a_n=\dfrac{1}{4}\cdot a_{n-1}}$