Explanation:
To find the recursive rule for a geometric sequence, we need to determine the common ratio, which is the constant ratio between any two consecutive terms.
In this case, let's consider the sequence 6, -18, 54, -162...
To find the common ratio, we can divide any term by its previous term. Let's take the second and first terms:
-18 / 6 = -3
We can see that the common ratio is -3.
Now, we can use this common ratio to write the recursive rule for the sequence. A recursive rule expresses each term in the sequence in terms of the previous term(s).
Let's denote the first term of the sequence as a₁, and the common ratio as r.
The recursive rule for a geometric sequence is:
aₙ = r * aₙ₋₁
In our case, the first term a₁ is 6, and the common ratio r is -3.
Therefore, the recursive rule for the given geometric sequence is:
aₙ = -3 * aₙ₋₁
This rule means that each term in the sequence is obtained by multiplying the previous term by -3. To find any term in the sequence, you would need to know the previous term and apply this rule.
For example, to find the 4th term (a₄), you would multiply the 3rd term (a₃) by -3:
a₄ = -3 * a₃ = -3 * 54 = -162
By applying the recursive rule, you can continue to find other terms in the sequence.