Questions: Which rule is a recursive rule for the sequence 1,-6,36,-216, ... ? an=6 cdot an-1 an=-6 cdot an-1 an=-1/6 cdot an-1 an=1/6 cdot an-1

Transcript text: Which rule is a recursive rule for the sequence $1,-6,36,-216, \ldots$ ? $a_{n}=6 \cdot a_{n-1}$ $a_{n}=-6 \cdot a_{n-1}$ $a_{n}=-\frac{1}{6} \cdot a_{n-1}$ $a_{n}=\frac{1}{6} \cdot a_{n-1}$

Solution

Solution Steps

To determine the recursive rule for the given sequence, we need to identify the pattern in the sequence. We can do this by examining the ratio between consecutive terms.

Solution Approach

Calculate the ratio between consecutive terms in the sequence.
Identify which of the given options matches the calculated ratio.

Step 1: Calculate the Ratios Between Consecutive Terms

Given the sequence $1, -6, 36, -216$, we calculate the ratios between consecutive terms: \[ \text{ratio}_1 = \frac{-6}{1} = -6.0000 \] \[ \text{ratio}_2 = \frac{36}{-6} = -6.0000 \] \[ \text{ratio}_3 = \frac{-216}{36} = -6.0000 \]

Step 2: Identify the Pattern

The calculated ratios are: \[ [-6.0000, -6.0000, -6.0000] \] Since all the ratios are $-6.0000$, we can identify the pattern.

Step 3: Determine the Recursive Rule

The recursive rule that matches the pattern is: \[ a_n = -6 \cdot a_{n-1} \]