Question

what is the recursive rule for this geometric sequence? 27, 9, 3, 1, ... enter your answers in the boxes. $a_n = \square \cdot a_{n - 1}$ $a_1 = \square$

Explanation:

Step1: Identify the first term

The sequence starts with 27, so \(a_1 = 27\).

Step2: Find the common ratio

To find the common ratio \(r\) in a geometric sequence, divide a term by its previous term. For example, \(\frac{9}{27}=\frac{1}{3}\), \(\frac{3}{9}=\frac{1}{3}\), \(\frac{1}{3}=\frac{1}{3}\). So the recursive formula is \(a_n = r \cdot a_{n - 1}\) where \(r=\frac{1}{3}\), so \(a_n=\frac{1}{3}\cdot a_{n - 1}\).

Answer:

For \(a_n=\square\cdot a_{n - 1}\), the value in the square is \(\frac{1}{3}\); for \(a_1=\square\), the value in the square is \(27\).